The clothing retailer, Gap, has apologised to the Chinese government for selling t-shirts featuring a map of China, minus South Tibet, Taiwan, and the South China Sea.
According to Gap, the t-shirts were distributed in "some overseas markets", and "mistakenly failed to reflect the correct map of China".
The error was noticed by a few discerning shoppers in Canada and posted online. The shirts have since been recalled and destroyed.
Of course, such an apology doesn't come without its controversy. Taiwan has been critical of Gap for siding with the Chinese government and is not the first international company to come under fire.
Air Canada, Zara, Marriott, Qantas, and Delta Air Lines have all issued an apology to China in recent months and have been accused by Taiwan for "buck[ling] under the pressure".
These apologies came after Chinese authorities reached out to various corporations noting that their websites appeared to list Taiwan as its own country.
China adamantly lays claim to Taiwan as an important part of their territory, while Taiwan's government, a pro-independence party, denies that this is the case.
This is a long-contested and sensitive issue since China and Taiwan separated in 1949 after a civil war which ended with a communist victory.
The Chinese government's active involvement in seeking out and confronting large corporations has been condemned by some.
"This is Orwellian nonsense and part of a growing trend by the Chinese Communist Party to impose its political views on American citizens and private companies".
However, the companies in question don't appear to share the sentiments and Gap certainly did all they could to appease Chinese concerns.


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I will run personal errands on behalf of my client's such as grocery shopping, pharmacy pick-up, and any other necessary personal errands that need to be done. Gas and mileage for trips up to 10 miles is at no extra charge and is included in the hourly rate. For distances driven on behalf of the client that exceeds 10 miles, or is outside of the city limits of their residence, a travel fee will be charged at the rate of $5.50/quarter hour, as well as, $.50 per mile.


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Support COVID-19 Research
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World AIDS Day: IAS launches new Global Scientific Research Strategy to cure HIV
Researchers continue to follow the science towards a scalable cure amid breakthrough cases such as the Esperanza patient
1 December 2021 (Geneva, Switzerland) – IAS – the International AIDS Society – has launched its latest strategy to find an HIV cure. Research Priorities for an HIV Cure: IAS Global Scientific Strategy 2021, was published today, World AIDS Day, in the journal, Nature Medicine.
The third edition of the IAS strategy comes soon after it was announced that an Argentinean woman, known as the ''Esperanza patient'', had been cured of HIV through natural immunity. This breakthrough has provided new hope for an HIV cure.
The curing of Timothy Ray Brown, also known as the "Berlin patient", of HIV in 2008 inspired the IAS to set up the Towards an HIV Cure programme, founded by HIV co-discoverer and Nobel Laureate Françoise Barré-Sinoussi, in 2010. The aim is to drive concerted efforts to accelerate global scientific research and engagement towards a cure for HIV.
At AIDS 2012, the 19th International AIDS Conference, in Washington D.C., the IAS launched the inaugural Towards an HIV cure: A global scientific strategy, which established cure as a pillar of the global HIV response. An updated strategy was released in 2016. The latest iteration comes after a decade of progress, which saw Adam Castillejo (the "London patient") cured via a stem cell transplant and two women seemingly cured through natural immunity. In addition, several interventions have shown great promise for a cure in animal models of HIV.
"It is clear that an HIV cure is achievable. The past five years of following the science have taught us that," Adeeba Kamarulzaman, President of the IAS and Director of the Centre of Excellence for Research in AIDS at the University of Malaya in Malaysia, said.
"The recent case of the cured Esperanza patient adds a new dimension to our understanding of HIV controllers, those people who naturally control HIV without antiretroviral therapy. The challenge is to identify how commonly elimination of all intact virus occurs in elite controllers and what the mechanism is that makes that possible. We can then work out how we can replicate it on a broad scale. We need an HIV cure that works for everyone so we can end the HIV pandemic.
"The surfacing of the Omicron SARS-CoV-2 variant is a reminder to us all that vaccine inequity and our failure to end infectious diseases, such as HIV and tuberculosis, are the same side of the coin – a devaluation of human life in poor countries.''
A 68-member IAS International Scientific Working Group developed the 2021 strategy over a year. The strategy reflects on progress made, identifies gaps in the research agenda, and outlines recommendations for HIV cure research over the next five years.
What we've learnt:
Several instances of remission and eradication have been recorded. Research towards an HIV cure is looking at these people, as well as HIV controllers, to better understand how a cure could work and help identify targets for HIV cure-related interventions.
Research on HIV controllers has provided new information and directions for cure research.
HIV reservoirs (cells that are the source of HIV once antiretroviral therapy, or ART, is stopped) are now seen as evolving rather than static sources of HIV not reached by ART. Important differences among people with HIV have been identified, such as biological sex, which affect reservoir location and dynamics.
Technological developments allow for a better understanding of how HIV latency is established and how the virus is reactivated after an analytical treatment interruption (conducted in the context of HIV cure research). This allows for the identification of new targets for HIV cures.
New drugs and interventions are being developed and have started to show success in animal (preclinical) models.
The key to a cure: the HIV reservoir
"Over the last decade, HIV cure research has greatly intensified but it remains clear that we will not cure HIV until we better understand where and how the virus hides. There has been substantial progress but more fundamental research is still needed," Professor Sharon Lewin, Co-Chair of the IAS Towards an HIV Cure programme, President-Elect of the IAS and Director of the Peter Doherty Institute for Infection and Immunity in Australia, said.
To date, numerous trials have been designed to shock and kill the virus. Those trials have confirmed just how difficult that task is and also prompted the strategy to incorporate research that uses diverse ways to respond to challenges in targeting the HIV reservoir:
"Poke and clear" (previously "shock and kill") involves drugs called latency-reversing agents (LRAs) to awaken the dormant virus in HIV-infected cells. Following activation, the virus-producing cells are eliminated by a second intervention.
"Block and lock" is a strategy on the opposite side of the spectrum that, instead of awakening the HIV reservoir, drives the reservoir into a deeper permanent resting state.
"Reduce and control" attempts to reduce the size of the reservoir and help the immune system control viral replication without the need for long-term ART.
These approaches may use one or more agents, including broadly neutralizing antibodies (bNAbs), various killer cells, therapeutic vaccines, and cell and gene therapies.
Beyond the science, the journey to an HIV cure will also require:
Coordinating investments to pursue the most promising HIV cure research approaches
Strengthening international collaborations to ensure a multidisciplinary approach for the development of a cure that is accessible and scalable in diverse settings
Promoting the involvement of early-career researchers and researchers from countries most affected by HIV
Supporting communication and information exchange between public and private sector researchers working on HIV cure research to alleviate the regulatory and logistical challenges associated with drug development
Building HIV cure research capacity among different populations and settings to generate increased HIV cure research and advocacy globally
Enhancing community engagement in HIV cure research through knowledge dissemination and capacity building for advocacy and effective engagement to represent their interests
Boosting research on the psychosocial and ethical implications of participating in HIV cure clinical research to ensure that both participants and clinical trial designers are well-informed and prepared.
Key research goals for the next five years:
Understanding HIV reservoirs
HIV reservoir measurement
Mechanisms of virus control
Targeting the provirus
Targeting the immune system
Cell and gene therapy
Paediatric remission and cure
Social, behavioural and ethical aspects of cure
About IAS – the International AIDS Society
IAS – the International AIDS Society – leads collective action on every front of the global HIV response through its membership base, scientific authority and convening power. Founded in 1988, the IAS is the world's largest association of HIV professionals, with members in more than 170 countries. Working with its members, the IAS advocates and drives urgent action to reduce the impact of HIV. The IAS is also the steward of the world's most prestigious HIV conferences: the International AIDS Conference, the IAS Conference on HIV Science, and the HIV Research for Prevention Conference.
For more information, visit www.iasociety.org.
About the IAS Towards an HIV Cure Programme
Established in 2010, the IAS Towards an HIV Cure programme drives concerted efforts to accelerate global scientific research and engagement towards a cure for HIV. Particular focus is given to advancing the HIV cure field in settings where resources for HIV cure research are limited and facilitating interaction with HIV and other biomedical research areas.
About the International AIDS Conference
The International AIDS Conference is the largest gathering on HIV and AIDS in the world. First convened during the peak of the AIDS epidemic in 1985, it continues to provide a unique forum for the intersection of science, advocacy and human rights. Each conference is an opportunity to strengthen policies and programmes that ensure an evidence-based response to the epidemic. The conference also serves as a focal point to intensify political and financial commitments to HIV and related epidemics. AIDS 2022, the 24th International AIDS Conference, will be held in Montreal, Canada, and virtually from 29 July to 2 August 2022.
For more information, visit www.aids2022.org.
For any media enquiries, please contact:
Rebecca Elliott
T: +61 (0) 3 8344 8360
M: +61 412 103 306
E: rebecca.elliott@unimelb.edu.au

redpajama_set_name: RedPajamaCommonCrawl


Not all superpowers are created equal. Being more than human can more of a burden than a gift, especially if your special power is, say, acid tears or bullet attraction.
A new Threadless tee celebrates the hilarious plight of the D-list hero with its new "Useless Superpowers" shirt, available for both men and women in Sinestro yellow.
There's also a whole blog devoted to archiving useless superpowers, and they've put out a call for submissions if you think have ideas counterproductive and impractical enough to make the cut.


text:
It's polo season
Polo NewsNorth America
Hickory Hall Polo Club will kick off its summer-long series of matches benefiting area charities, "Polo at Sunset" this week.
Starting Friday and continuing each week, the club will play host and fundraise each Friday for a different cause. Additionally, two Saturday (noon start) events will be held under the name "Polo on the Prairie" for those who prefer a daytime game.
The sport, which has been quietly present in Boone County for decades, has become a favorite outing for area families. What was traditionally an event of formal attire, stuffed shirts, and stemmed wine glasses, has morphed into a great evening to spend with family, friends, and neighbors.
Some folks do come prepared with tents, fine china, and champagne – just for the fun of living out the polo scenes from the movie "Pretty Woman." Others pitch a picnic blanket, and stuff a cooler with submarine sandwiches, sodas and beer. Whether you choose cargo shorts and your favorite T-shirt, or a seersucker suit and Kentucky Derby-worthy fascinator hat, you're guaranteed to fit right in.
On Fridays, gates open at 5 p.m. and play begins at 6:15 p.m. The general admission of $20 per carload keeps the price point of the event within reach of most families. Those attending are always encouraged to bring refreshments and other creature comforts like bug repellent, sunscreen and chairs.
Although there are no onsite concessions, restroom and handwashing facilities are available. However some showcased charities do choose to co-op with local favorites like LA Cafe for their events. During these dates, patrons can order boxed meals ahead of time (but be sure to check the schedule and website).
While checking the website, look for special reserved premium tailgating spots available during select events starting at $250 with room for six guests at a table. Be sure to bring a little cash if you would like to participate in any of the additional side games available. Each charity tends to do something a bit different, but most will have opportunities for guests to take part in raffles, wine pulls, and other fun add-ons.
Families are also always welcome to pay the $20 carload fee at the gate and make your own fun at no additional cost.
Many evenings also feature a low swooping airplane flyover. This special treat is always a big hit with the younger set, since the planes drop candies as they pass at halftime. Adding to the fun is the crowd favorite and time honored practice of the halftime Divot Stomp. All guests are invited onto the field. They then locate any kicked up sod and give it a solid "stomp" to put divots back into their place. Once the playing surface is sufficiently repaired by spectators, the game is set to resume.
If you're thinking polo is something relatively new to Central Indiana, you may be surprised to know that it was first brought to the area more than 100 years ago. One of the fathers of the Indianapolis Motor Speedway, Carl Fisher, introduced his friends to the high adrenaline sport on the lawn of his estate along Cold Spring Road (now a part of the Marian University campus).
Today, polo in the metro area is a family affair. The Greg Chandler family now owns and operates Indy Polo as Hickory Hall Polo Club. Greg fell in love with the sport while working at the nearby (now defunct) Rancho Alegre, where the Longwood facility owned by his father Tom in partnership with Fred Stauder was located. That storied ranch, stables, and play field is now just a memory covered over by a subdivision at Zionsville's north end.
Chandler's son, Austin, now serves as the club president and sees to the day to day details of the facility, while his wife Shannon and Audra Poe tend to the media and charitable interface duties. Everyone has a day job though – Hickory Hall is a passion project, not a get rich quick scheme.
Those who attend matches this summer will see a variety of participants on mount. Women compete side-by-side with men, and the age range is highschoolers through "seasoned" participants. When asked about the toll this sport takes on backs, Austin Chandler laughed, "It's not a matter of if you'll break something, it's a matter of when and what."
He knows quite well the hazards of play, having suffered a broken back a few years ago. He points to one of the guys who practically glides from the ground to the saddle with a fluid motion. "He's been playing for 37 years."
A total of 19 non-profits will benefit from the Friday and Saturday games this year. Those of particular interest to Boone County residents include the Boys & Girls Club, zWorks, Youth and High School Rugby of Indiana, Progress House by Aspire, Special Olympics, and the season finale on Oct. 2 benefitting the Jacob Pickett Response Organization.
Zionsville based zWorks will again partner with the Boys & Girls Club of Boone County for their event at Hickory Hall this year.
"Last year was very successful with a record turnout," notes zWorks Executive Director Vickie Hall. Katie Reasoner, who serves as resource director at the B&GCBC, agrees, adding "We look forward to seeing everyone at the polo field in Whitestown for a great evening that supports our missions and our community."
Hickory Hall has hosted charities for eight years now. Over this time, they've helped these worthy causes raise more than $2 million. Last year alone brought in more than $350,000.
For up to date information, to find out where to buy tickets and reserved tailgate spots, or to sign up to take polo lessons, see Hickory Hall's website at www.IndyPolo.com The club is at 7551 E. 100 N., Whitestown, just a few miles north of Zionsville.


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Last updated on November 25, 2009 January 21, 2014 by Richard N Williams for Armagard Ltd.
Digital signage is a relatively new concept although the idea of using TV and display systems outside the home has been around for a while but it wasn't until the development of flat panel thin film transistor (TFT) TVs such as LCD or plasma screens that digital signage really took off.
Now there doesn't seem to be any place where an LCD or plasma screen isn't relaying a marketing message or providing information; shopping malls, high streets, schools, colleges and even hospitals are all benefiting from the use of these TV displays as digital signage systems.
And using televisions for digital out of home advertising (Dooh) continues to increase as more and more businesses are turning to digital signage and more and more locations are being found to site TV screens for advertising and information.
And one of these burgeoning locations is using these TV screens outside for outdoor digital signage or even outdoor entertainment in bars, restaurants and pubs.
The only problem with using outdoor TV systems and digital outdoor signage is that the LCD or plasma TV needs to be protected from the elements, in particular the rain, wind, heat and cold temperatures.
LCD enclosures are a flexible and cost effective method of allowing standard TV systems outside in the elements. Not only do they offer NEMA and IP rated waterproofing but they also contain additional heating and air flow systems to ensure the enclosed LCD or plasma is kept at perfect operating temperatures.
LCD enclosures also have additional protection for te LCD screen by utilising anti-shatter polycarbonates ensuring the TV system isn't damaged even by deliberate attempts.
This entry is filed under digital signage, IP rating, lcd enclosure, outdoor advertising. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.


Joshua Becker of Becoming Minimalist shares lessons learned from intentionally letting go.
Episode 629: Lessons Learned from Intentionally Letting Go by Joshua Becker of Becoming Minimalist (Mindful Living & Simplicity).
Mary Jaksch of Good Life Zen shares how to reset your emotional thermostat.
Episode 628: How to Reset Your Emotional Thermostat For a Happier Life by Mary Jaksch of Good Life Zen (Mindfulness & Happiness).
Marc Chernoff of Marc and Angel Hack Life shares 7 habits that are making you miserable.
Episode 627: 7 Habits That Are Making You Miserable by Marc Chernoff of Marc & Angel Hack Life (Mindfulness & Happiness).
David Cain guest-posts on Cait Flanders' site with a theory: maybe we aren't materialistic enough.
Episode 626: We Are Not Materialistic Enough by David Cain with Cait Flanders (Simple Living & Minimalism).
Courtney Carver of Be More With Less shares 10 simple ways to worry less.
Episode 625: 10 Simple Ways to Worry Less by Courtney Carver of Be More With Less (Mindfulness & Happiness).
Ramit Sethi shares the importance of knowing when to quit.
Episode 624: When It's Time to Walk Away by Ramit Sethi of I Will Teach You to be Rich (Developing Habits & Personal Growth).
Colin Wright shares an excerpt from his book Come Back Frayed.
Episode 623: More Sense in Motion by Colin Wright of Exile Lifestyle (Simple Living & Minimalism).
Leo Babauta of Zen Habits shares how and why you should take time to reflect.
Episode 622: 5 Powerful Reasons to Make Reflection a Daily Habit & How To Do It by Leo Babauta of Zen Habits (Mindful Living & Simplicity).
Allison Fallon with No Sidebar talks about the importance of decisions in your life.
Episode 621: Are You Making Good Decisions by Allison Fallon with No Sidebar (Developing Habits & Personal Growth).
Steve Pavlina shares why you have the right to be wrong.
Episode 620: You Have the Right to be Wrong by Steve Pavlina (Developing Habits & Personal Growth).
Guest author Dori Cameron with Becoming Minimalist shares 5 myths that encourage excess spending.
Episode 619: 5 Myths That Encourage Excess Spending by Dori Cameron with Becoming Minimalist (Saving Money & Debt Free).
Mary Jaksch of Good Life Zen shares 3 strategies to boost confidence and feel good about yourself.
Episode 618: 3 Strategies to Boost Confidence & Feel Good About Yourself by Mary Jaksch of Good Life Zen (Social Psychology & Health).
Marc Chernoff of Marc and Angel Hack Life shares how to step forward when you're scared to death.
Episode 617: 5 Ways to Step Forward When You're Scared to Death by Marc Chernoff - Marc and Angel Hack Life (Tips to Face Your Fears).
Cal Newport of the Study Hacks blog shares what weakest link theory is and how it applies to time management.
Episode 616: Monday Master Class: Apply Weakest Link Theory to Time Management by Cal Newport (Productivity & Efficiency).
Leo Babauta of Zen Habits shares why we have crutches and how to let go of them.
Episode 615: Simplify: Let Go of Your Crutches by Leo Babauta of Zen Habits (Simple Living & Minimalism).
Krissy McNeill with No Sidebar shares a story of slowing down after stopping in a store.
Episode 614: Journeying Into a Slower Lifestyle by Krissy McNeill with No Sidebar (Simple Living & Minimalism).
Steve Pavlina shares the best way to overcome the fear of public speaking.
Episode 613: Overcoming Fear of Public Speaking by Steve Pavlina (Confidence & Self Esteem).
Joshua Becker of Becoming Minimalist shares a helpful guide for decluttering toys.
Episode 612: A Helpful Guide for Decluttering Toys from Becoming Minimalist (Simple Living & Minimalism).
Ramit Sethi of I Will Teach You to be Rich shares some simple steps to overcome social anxiety (Part 2 of 2).
Episode 611: How to Overcome Social Anxiety - Part 2 by Ramit Sethi (Developing Habits & Personal Growth).
Ramit Sethi of I Will Teach You to be Rich shares some simple steps to overcome social anxiety (Part 1 of 2).
Episode 610: How to Overcome Social Anxiety - Part 1 by Ramit Sethi of I Will Teach You to be Rich (Confidence).
Mary Jaksch of Good Life Zen shares different ways to approach persistent thoughts in meditation.
Episode 609: How to Deal with Pesky Thoughts in Meditation by Mary Jaksch of Good Life Zen (Mindful Living & Simplicity).
Marc Chernoff of Marc and Angel Hack Life shares 20 things that will matter a lot less to you in 20 years (part 2 of 2).
Episode 608: 20 Things That Will Matter a Lot Less To You in 20 Years - Part 2 by Marc Chernoff of Marc And Angel (Mindful Living & Simplicity).
Marc Chernoff of Marc and Angel Hack Life shares 20 things that will matter a lot less to you in 20 years (part 1 of 2).
Episode 607: 20 Things That Will Matter a Lot Less To You in 20 Years - Part 1 by Marc Chernoff of Marc and Angel Hack Life (Stay Calm).
Cal Newport asks why we never model our actions after those who have succeeded before us, and offers his opinion on it.
Episode 606: Do What Works, Not What's Satisfying: Pseudo-Striving and our Fear of Reality-Based Planning by Cal Newport (Learning).
Leo Babauta of Zen Habits describes different ways to deal with family, friends, and roommates who do not want to simplify or minimize their lives with you.
Episode 605: How to Find Peace Living With a Packrat by Leo Babauta of Zen Habits (Minimalism with Family, Friends, and Roommates).


text:
\section{Introduction}

Away from equilibrium, currents of energy and matter flow across open systems.  Because of the atomic structure of matter, these currents manifest fluctuations, which can be characterized by their large-deviation properties within the framework of probability theory.  Furthermore, the motions of microscopic particles obey fundamental time-reversal symmetry, which implies that the current fluctuations satisfy the so-called fluctuation theorems \cite{ECM93,G96,K98,LS99,J11,G04JCP,AG04,AG06,DDR04,D07,SIPGD07,PGPD10,PGLD13,G13NJP,BDGJL15}.  These fluctuation theorems are usually obtained in the long-time limit.  Remarkably, fluctuation theorems may also hold at every finite time under specific conditions.  This is the case for Markov jump processes describing linear reactions in homogeneous systems \cite{AG08}.  Recently, a finite-time fluctuation theorem was established for diffusion-influenced surface reactions in spatially extended systems described by stochastic partial differential equations \cite{TDFT}.

Fluctuation theorems play a key role in the determination of the thermodynamic forces driving systems into nonequilibrium steady states.  Under such circumstances, the measurement of the driving forces is carried out in the long-time limit.  In systems where a finite-time fluctuation theorem holds, the driving forces are defined at every finite time, and this allows one to investigate how they vary in time and to determine the time scale on which they  converge to their asymptotic values.

The purpose of this paper is to investigate aspects of the finite-time fluctuation theorem for the diffusion-influenced surface reaction ${\rm A}\rightleftharpoons{\rm B}$ on spherical and Janus catalytic particles.  We deduce analytical expressions for the time dependence of the thermodynamic force of the reactive process by solving macroscopic diffusion equations with the special boundary conditions obtained in Ref.~\cite{TDFT}. This solution provides the large-deviation properties of the random number of reactive events occurring during finite-time intervals.  The theoretical results are compared with numerical simulations carried out using two different methods.  In the first method, the process is simulated with particles carrying a color (A or B) and independently diffusing according to a random walk algorithm between the reservoir and the catalytic surface where they may interchange their color.  In the second method, the simulation is performed using the multiparticle collision dynamics \cite{K08}.  In this method, the system comprises reactive and solvent particles, together with a spherical composite catalytic particle made from a collection of linked beads \cite{HSGK18}, some of which catalyze the reaction ${\rm A}\rightleftharpoons{\rm B}$.  Since the catalytic particle is larger than the reacting and solvent particles, its motion is small and negligible, allowing us to make comparisons with the theory that assumes immobile catalytic surfaces.

The paper is organized as follows.  The principal results of the finite-time fluctuation theorem are summarized in section~\ref{Sec-thm}.  The theory is applied to a spherical catalytic particle in section~\ref{S-geom}, and a Janus catalytic particle in section~\ref{J-geom}.  In these sections, the deterministic diffusion equations are solved for the geometries of the particles in these systems.  Complete analytic expressions are obtained for the time dependence of the rates and the corresponding affinity.  For each system, theory is compared with numerical simulations.  Section~\ref{conclusion} gives concluding remarks and perspectives.

\section{The finite-time fluctuation theorem}
\label{Sec-thm}

We consider a system where the molecular species A and B diffuse in a three-dimensional domain $V$ extending between three surfaces $\partial V = S_{\rm cat}\cup S_{\rm inert}\cup S_{\rm res}$.  The reaction ${\rm A}\rightleftharpoons{\rm B}$ takes place at the catalytic surface $S_{\rm cat}$.  The molecules ${\rm A}$ and ${\rm B}$ are reflected at the inert surface $S_{\rm inert}$.  Moreover, they enter and exit the domain $V$ at the surface $S_{\rm res}$ in contact with a reservoir.  Accordingly, the concentrations (i.e., the densities) $c_k$ of the species $k\in\{{\rm A},{\rm B}\}$ are ruled by the fluctuating diffusion equations
\begin{equation}
\partial_t \, c_k + \pmb{\nabla}\cdot {\bf j}_k = 0 \, ,\qquad\mbox{with}\qquad  {\bf j}_k  = -D_k \pmb{\nabla} c_k + \pmb{\eta}_k\, , \label{diff-eq-k}
\end{equation}
and boundary conditions
\begin{eqnarray}
{\rm if}\ {\bf r}\in S_{\rm cat} : \qquad  && D_{\rm A} \, \partial_{\bot} c_{\rm A}({\bf r},t) = -D_{\rm B}\, \partial_{\bot}c_{\rm B}({\bf r},t) =\kappa_+ c_{\rm A}({\bf r},t) - \kappa_- c_{\rm B}({\bf r},t)+ \xi({\bf r},t) \, , \label{bc-cat}\\
{\rm if}\ {\bf r}\in S_{\rm inert} : \qquad && \partial_{\bot} c_k({\bf r},t) = 0 \, , \label{bc-inert}\\
{\rm if}\ {\bf r}\in S_{\rm res} : \qquad && c_k({\bf r},t) = \bar{c}_k \, ,  \label{bc-res}
\end{eqnarray}
where  $D_k$ are the diffusion coefficients, $\partial_{\bot}$ is the gradient in the direction normal to the surface and oriented towards the interior of the domain~$V$, $\kappa_{\pm}$ are the surface reaction rate constants, and $\bar{c}_k$ the concentration values at the reservoir. The Gaussian noises associated with bulk diffusion are characterized by
\begin{equation}
\langle\pmb{\eta}_k({\bf r},t)\rangle = 0 \, , \qquad \langle\pmb{\eta}_k({\bf r},t)\otimes\pmb{\eta}_{k'}({\bf r'},t')\rangle = 2\,  D_k \, c_k({\bf r}, t) \, \delta_{kk'} \, \delta({\bf r}-{\bf r'}) \, \delta(t-t') \, {\boldsymbol{\mathsf 1}} \, ,
\end{equation}
where $k,k'\in\{{\rm A},{\rm B}\}$ and ${\boldsymbol{\mathsf 1}}$ is the $3\times 3$ identity matrix, while the Gaussian noise associated with surface reaction satisfies
\begin{equation}
\langle \xi({\bf r},t) \rangle = 0 \, , \qquad
\delta^{\rm s}({\bf r})\, \langle \xi({\bf r},t)\,\xi({\bf r'},t') \rangle\, \delta^{\rm s}({\bf r'}) = (\kappa_+\, c_{\rm A}+\kappa_-\, c_{\rm B}) \, \delta^{\rm s}({\bf r}) \, \delta({\bf r}-{\bf r'}) \, \delta(t-t')  \, ,
\end{equation}
which is expressed in terms of surface delta distributions $\delta^{\rm s}({\bf r})$ nonvanishing if ${\bf r}\in S_{\rm cat}$ \cite{BAM76}.

As proved in Ref.~\cite{TDFT}, the probability $P(n,t)$ that the net number $n$ of reactive events ${\rm A}\to{\rm B}$ have occurred during the time interval $[0,t]$ satisfies the finite-time fluctuation theorem
\begin{equation}
\boxed{\frac{P(n,t)}{P(-n,t)} = \exp ( {\cal A}_t \, n )}
\label{FT}
\end{equation}
{\it at every time}.  The time-dependent affinity is defined as
\begin{equation}
{\cal A}_t = \ln\frac{W_t^{(+)}}{W_t^{(-)}}
\label{A-dfn}
\end{equation}
with the finite-time rates
\begin{equation}
W_t^{(\pm)} = W_{\infty}^{(\pm)} + \frac{1}{t}\, \Psi(t) \, , \label{W}
\end{equation}
having the asymptotic values
\begin{equation}
W_{\infty}^{(+)} = \Sigma \kappa_+ \bar{c}_{\rm A} \,  , \qquad\qquad
W_{\infty}^{(-)} = \Sigma \kappa_- \bar{c}_{\rm B} \, , \label{W-inf}
\end{equation}
and a time dependence given by the function
\begin{equation}
\Psi(t) = \ell^2 \kappa_+\kappa_- \left[ \frac{\bar{c}_{\rm B}}{D_{\rm A}^2} \Upsilon_{\rm A}(t) + \frac{\bar{c}_{\rm A}}{D_{\rm B}^2} \Upsilon_{\rm B}(t)\right] .
\label{Psi}
\end{equation}
The asymptotic values~(\ref{W-inf}) are expressed in terms of the effective catalytic surface area
\begin{equation}
\Sigma = \int_{\rm cat} dS \, (1-\phi) \, ,
\label{Sigma}
\end{equation}
where $\phi$ is the solution of the Laplace equation,
\begin{equation}
\nabla^2 \phi = 0 \, , \label{phi1}
\end{equation}
with the boundary conditions
\begin{equation}
\left(\partial_{\bot} \phi\right)_{\rm cat} = \ell^{-1}(\phi-1)_{\rm cat} \, , \qquad \left(\partial_{\bot} \phi\right)_{\rm inert} = 0 \, , \qquad\mbox{and}\qquad \left(\phi\right)_{\rm res} = 0 \, , \label{phi4}
\end{equation}
involving the characteristic length of the diffusion-influenced surface reaction,
\begin{equation}
\ell \equiv \left(\frac{\kappa_+}{D_{\rm A}} + \frac{\kappa_-}{D_{\rm B}}\right)^{-1} .
\label{ell}
\end{equation}
The stationary concentration fields are thus given for $k={\rm A},{\rm B}$ by
\begin{equation}
\langle c_k\rangle_{\rm st} = \bar{c}_k +\frac{\nu_k\ell}{D_k}   \left(\kappa_+\bar{c}_{\rm A}-\kappa_- \bar{c}_{\rm B}\right)  \phi \, ,
\label{NESS}
\end{equation}
where $\nu_{\rm A}=-1$ and $\nu_{\rm B}=+1$ are the stoichiometric coefficients of the surface reaction.

Equation~(\ref{Psi}) is written in terms of the functions
\begin{equation}
\Upsilon_k(t) = \int dV \phi({\bf r})\left[ \phi({\bf r})-f_k({\bf r},t)\right] \, ,
\label{Upsilon}
\end{equation}
where $f_k$ is the solution of the diffusion equations,
\begin{equation}
\partial_t f_k = D_k \nabla^2 f_k \, , \label{f1}
\end{equation}
with the boundary and initial conditions
\begin{equation}
\left(\partial_{\bot} f_k\right)_{\rm cat} = \left(\frac{\kappa_+}{D_{\rm A}}\, f_{\rm A}+\frac{\kappa_-}{D_{\rm B}}\, f_{\rm B}\right)_{\rm cat} \, , \qquad \left(\partial_{\bot} f_k\right)_{\rm inert} = 0 \, , \qquad
\left(f_k\right)_{\rm res} = 0 \, , \qquad \left(f_k\right)_{t=0} = \phi \, , \label{f5}
\end{equation}
for $k={\rm A},{\rm B}$.

\begin{figure}[h]
\centerline{\scalebox{0.42}{\includegraphics{fig1.eps}}}
\caption{Schematic representation of the probability distributions $P(\pm n,t)$ of opposite fluctuations in the number $n$ of reactive events occurring during the time interval $[0,t]$ under nonequilibrium conditions.  These distributions shift away from the origin $n=0$ as time $t$ increases, so that their overlap rapidly decreases.}
\label{fig1}
\end{figure}

The mean current, i.e., the mean overall reaction rate, is given by
\begin{equation}
{\cal J}  = W_t^{(+)} - W_t^{(-)} = \Sigma\left( \kappa_{+}\bar{c}_{\rm A} - \kappa_{-}\bar{c}_{\rm B} \right) \, , \label{J}
\end{equation}
which does not depend on time because of the expression~(\ref{W}) for the rates; however, the diffusivity of the current fluctuations
\begin{equation}
{\cal D}_t  = \frac{1}{2} \left(W_t^{(+)} +W_t^{(-)}\right) \label{D_t}
\end{equation}
does depend on time.

We note that the rates~(\ref{W}) are determined at early times by the stationary solutions~(\ref{NESS}) according to
\begin{eqnarray}
&& W_t^{(+)} = \int_{\rm cat} dS \, \kappa_+\langle c_{\rm A}\rangle_{\rm st} + O(t) \, , \label{W+early}\\
&& W_t^{(+)} = \int_{\rm cat} dS \, \kappa_- \langle c_{\rm B}\rangle_{\rm st} +O(t) \, . \label{W-early}
\end{eqnarray}

As schematically depicted in figure~\ref{fig1}, the probability distributions $P(\pm n,t)$ shift in opposite directions under nonequilibrium conditions, because their mean values are given by $\langle n\rangle_t=\pm{\cal J}t$ in terms of the mean current~(\ref{J}).  If the concentrations at the reservoir satisfy the equilibrium condition $\kappa_{+}\bar{c}_{\rm A} = \kappa_{-}\bar{c}_{\rm B}$, the time-dependent rates~(\ref{W}) are equal and the mean current~(\ref{J}) vanishes together with the affinity~(\ref{A-dfn}).  Detailed balance is thus recovered at equilibrium.

In the following sections, these results, proved in Ref.~\cite{TDFT}, are applied to spherical and Janus catalytic particles.

\section{Spherical catalytic particle}
\label{S-geom}

\subsection{Theory}
\label{S-geom-theory}

We consider a spherical catalyst of radius $r=R$ centered on the origin of the reference frame, and a reservoir for species A and B that is located at a sphere of radius $r=L\gg R$, $r$ denoting the radial distance from the origin.  For this spherical geometry, the Laplacian operator in the problem~(\ref{phi1})-(\ref{phi4}) for the stationary solution reduces to
\begin{equation}
\nabla^2\phi =\frac{1}{r}\frac{d^2}{dr^2}\left(r\, \phi \right) = 0 \, ,
\end{equation}
while $\partial_{\bot}=d/dr$.  Hence, the equations can be solved by setting $\phi(r)=u(r)/r$, yielding the solution
\begin{equation}
\phi(r) = R\, \frac{\rm Da}{\tilde\Delta} \left(\frac{1}{r}-\frac{1}{L}\right) ,
\label{phi}
\end{equation}
where
\begin{equation}
{\rm Da}\equiv \frac{R}{\ell} = R \left(\frac{\kappa_+}{D_{\rm A}} + \frac{\kappa_-}{D_{\rm B}}\right)
\label{Damkohler}
\end{equation}
is the Damk\"ohler number defined in terms of the characteristic length~(\ref{ell}), and
\begin{equation}
\tilde\Delta \equiv 1 + {\rm Da} \left(1 - \frac{R}{L}\right) .
\label{tilde-Delta-3D}
\end{equation}
The effective catalytic surface area~(\ref{Sigma}) is given by $\Sigma = 4\pi R^2/\tilde\Delta = 4\pi R^2/[1 + {\rm Da}\,(1 - R/L)]$.
Setting $f_k(r,t)=v_k(r,t)/r$, the functions~(\ref{Upsilon}) can be expressed as
\begin{equation} \label{up}
\Upsilon_k(t) =4\pi \int_R^L dr \, u(r) \left[ u(r)-v_k(r,t)\right] .
\end{equation}
The solutions of equations~(\ref{f1})-(\ref{f5}) can be expanded as
\begin{equation}
v_k(r,t) = \sum_{i=1}^{\infty} a_i \, {\rm e}^{-D_k q_i^2 t} \sin q_i(L-r) + \chi_k \sum_{i=1}^{\infty} \tilde a_i \, {\rm e}^{-D_k \tilde q_i^2 t} \sin \tilde q_i(L-r) \label{vk-3D}
\end{equation}
with the coefficients $\chi_{\rm A} = D_{\rm A}/\kappa_+$ and $\chi_{\rm B} = -D_{\rm B}/\kappa_-$
and where the eigenvalues are the roots of
\begin{eqnarray}
&& q_i R = -\Delta \, \tan q_i (L-R) \, , \label{eigen-3D}\\
&& \tilde q_i R = -\tan \tilde q_i (L-R) \, ,
\end{eqnarray}
with $\Delta \equiv 1 + {\rm Da} = \lim_{L\to\infty}\tilde\Delta$ and $q_i,\tilde q_i>0$.  Since the solutions~(\ref{vk-3D}) satisfy the initial conditions $v_k(r,t=0)=u(r)$ for $k={\rm A}\,,{\rm B}$, we find that the expansion coefficients are given by
\begin{equation}
a_i = \frac{4 R^2\left[ \sin q_i (L-R) - q_i(L-R) \cos q_i (L-R)\right]}{q_i L\ell \tilde\Delta \left[2q_i(L-R)-\sin 2q_i (L-R)\right]}
\end{equation}
and $\tilde a_i=0$.  Using equation~(\ref{eigen-3D}), the functions~(\ref{Upsilon}) are obtained as
\begin{equation}
\Upsilon_k(t) = \frac{8\pi R^2}{\ell^2(L-R)} \sum_{i=1}^{\infty} \frac{1-{\rm e}^{-D_k q_i^2t}}{q_i^2\left[q_i^2+\frac{\Delta\tilde\Delta}{R^2(1-R/L)}\right]} .
\end{equation}

Therefore, the rates are given by equation~(\ref{W}) with
\begin{equation}
W_{\infty}^{(+)} = \frac{4\pi R^2}{\tilde\Delta} \, \kappa_+ \bar{c}_{\rm A} \, , \qquad\qquad
W_{\infty}^{(-)} = \frac{4\pi R^2}{\tilde\Delta} \, \kappa_- \bar{c}_{\rm B} \,  ,
\label{W-inf-3D}
\end{equation}
and
\begin{equation}
\Psi(t) = \frac{4\pi R^5\kappa_+\kappa_-}{\left(\Delta\tilde\Delta\right)^{3/2}} \left(1-\frac{R}{L}\right)^{3/2}\left[ \frac{\bar{c}_{\rm B}}{D_{\rm A}^2}\, \Omega_{\rm s}\left(\gamma_{\rm A}t\right) + \frac{\bar{c}_{\rm A}}{D_{\rm B}^2}\, \Omega_{\rm s}\left(\gamma_{\rm B}t\right)\right]
\label{Psi-3D}
\end{equation}
with the function
\begin{equation}
\Omega_{\rm s}(\tau) \equiv \frac{2R}{L\sqrt{\Delta\tilde\Delta(1-R/L)}} \sum_{i=1}^{\infty} \frac{1-{\rm e}^{-Q_i^2\tau}}{Q_i^2\left(Q_i^2+1\right)}
\label{Omega-s}
\end{equation}
obtained after the substituting $Q_i=q_iR\sqrt{(L-R)/(L\Delta\tilde\Delta)}$ and defining the rates
\begin{equation}
\gamma_k \equiv \frac{D_k\Delta\tilde\Delta}{R^2\left(1-R/L\right)} \, .
\end{equation}
Using these results, $\Psi(t)$ can be determined by numerical evaluation of the sums.

Using the long-time limit of equations~(\ref{up}) and (\ref{vk-3D}) we find that the function~(\ref{Psi}) is given by
\begin{equation}
\Psi({\infty}) = \frac{4\pi R^4 L \kappa_+\kappa_-}{3\tilde\Delta^2} \left(1-\frac{R}{L}\right)^3 \left(\frac{\bar{c}_{\rm A}}{D_{\rm B}^2} + \frac{\bar{c}_{\rm B}}{D_{\rm A}^2} \right) .
\label{Psi-3D-t_inf}
\end{equation}
We see that this asymptotic value is proportional to the distance $L$ to the reservoirs and thus diverges in the limit $L\to\infty$.  Comparing equation~(\ref{Psi-3D-t_inf}) with the limit $t\to\infty$ of equation~(\ref{Psi-3D}), we get
\begin{equation}
\Omega_{\rm s}(\infty) = \frac{L\Delta^{3/2}}{3R\tilde\Delta^{1/2}} \left(1-\frac{R}{L}\right)^{3/2} \, .
\end{equation}
Consequently, on the long time scale $D_k^{-1}L^2 \ll t$ for $k\in\{{\rm A},{\rm B}\}$,
the finite-time affinity behaves as
\begin{equation}
{\cal A}_t = \ln\frac{\kappa_+\bar{c}_{\rm A} +\frac{1}{t} \frac{R^2L\kappa_+\kappa_-}{3\tilde\Delta}\left(1-\frac{R}{L}\right)^3\left(\frac{\bar{c}_{\rm A}}{D_{\rm B}^2}+\frac{\bar{c}_{\rm B}}{D_{\rm A}^2}\right)+O\left(t^{-1}{\rm e}^{-D_{\rm A}q_1^2t}\right) + O\left(t^{-1}{\rm e}^{-D_{\rm B}q_1^2t}\right)}{\kappa_-\bar{c}_{\rm B} +\frac{1}{t} \frac{R^2L\kappa_+\kappa_-}{3\tilde\Delta}\left(1-\frac{R}{L}\right)^3\left(\frac{\bar{c}_{\rm A}}{D_{\rm B}^2}+\frac{\bar{c}_{\rm B}}{D_{\rm A}^2}\right)+O\left(t^{-1}{\rm e}^{-D_{\rm A}q_1^2t}\right) + O\left(t^{-1}{\rm e}^{-D_{\rm B}q_1^2t}\right)} \, .
\end{equation}

If $L\gg R$, the rates~(\ref{W}) have the well-defined asymptotic values
\begin{equation}
W_{\infty}^{(+)} = \frac{4\pi R^2}{1+{\rm Da}} \, \kappa_+ \bar{c}_{\rm A} \, , \qquad\qquad
W_{\infty}^{(-)} = \frac{4\pi R^2}{1+{\rm Da}} \,  \kappa_- \bar{c}_{\rm B} \, ,
\label{W-inf-3D-lim}
\end{equation}
and
\begin{equation}
\Psi(t) = \frac{4\pi R^5\kappa_+\kappa_-}{\left(1+{\rm Da}\right)^3} \left[ \frac{\bar{c}_{\rm B}}{D_{\rm A}^2} \, \Omega\left(\gamma_{\rm A}t\right) + \frac{\bar{c}_{\rm A}}{D_{\rm B}^2}\, \Omega\left(\gamma_{\rm B}t\right)\right] ,
\label{Psi-3D-lim}
\end{equation}
where $\gamma_k = D_kR^{-2}( 1+{\rm Da})^2$
and the function $\Omega_{\rm s}(\tau)$ is given by its integral approximation,
\begin{equation}
\Omega(\tau) = \frac{1}{\pi} \int_{-\infty}^{+\infty} \frac{1-{\rm e}^{-Q^2 \tau}}{Q^2 \left(1 + Q^2\right)}\, dQ = 2\sqrt{\frac{\tau}{\pi}} -1 + {\rm e}^{\tau}\, {\rm erfc}(\sqrt{\tau})\, .
\label{Omega}
\end{equation}
This function has the following asymptotic expansion for $\tau\to\infty$
\begin{equation}
\Omega(\tau) = 2\sqrt{\frac{\tau}{\pi}} -1 + \frac{1}{\sqrt{\pi\tau}} + O\left(\frac{1}{\tau^{3/2}}\right) \, ,
\label{Omega-asympt}
\end{equation}
and Taylor series around $\tau=0$
\begin{equation}
\Omega(\tau) = \tau - \frac{4}{3\sqrt{\pi}}\, \tau^{3/2} + \frac{1}{2}\, \tau^2 - \frac{8}{15\sqrt{\pi}}\, \tau^{5/2} +O(\tau^3) \, .
\label{Omega-series}
\end{equation}
Consequently, the function~(\ref{Psi-3D-lim}) increases without limit as $\sqrt{t}$.  However, it is divided by the time $t$ in the expression~(\ref{W}) for the rates, which thus have the well-defined values~(\ref{W-inf-3D-lim}) in the long-time limit.

On the intermediate time scale $D_k^{-1}R^2/(1+{\rm Da})^2 \ll t \ll D_k^{-1}L^2$, the rates have the following expressions
\begin{equation}
W_t^{(\pm)} = W_{\infty}^{(\pm)} + \frac{2}{\sqrt{\pi t}} \frac{4\pi R^4 \kappa_+\kappa_-}{(1+{\rm Da})^2} \left(\frac{\bar{c}_{\rm A}}{D_{\rm B}^{3/2}}+\frac{\bar{c}_{\rm B}}{D_{\rm A}^{3/2}}\right) +O(t^{-1})
\label{Q-long-3D}
\end{equation}
in terms of their asymptotic values~(\ref{W-inf-3D-lim}).
Accordingly, the affinity behaves as
\begin{equation}
{\cal A}_t = \ln\frac{\kappa_+\bar{c}_{\rm A}+\frac{2}{\sqrt{\pi t}}\frac{R^2\kappa_+\kappa_-}{1+{\rm Da}}\left(\frac{\bar{c}_{\rm A}}{D_{\rm B}^{3/2}}+\frac{\bar{c}_{\rm B}}{D_{\rm A}^{3/2}}\right)+O(t^{-1})}{\kappa_-\bar{c}_{\rm B}+\frac{2}{\sqrt{\pi t}}\frac{R^2\kappa_+\kappa_-}{1+{\rm Da}}\left(\frac{\bar{c}_{\rm A}}{D_{\rm B}^{3/2}}+\frac{\bar{c}_{\rm B}}{D_{\rm A}^{3/2}}\right)+O(t^{-1})} .
\label{Aff-inter-3D}
\end{equation}
In the long-time limit, the affinity~(\ref{A-dfn}) thus converges towards the expected asymptotic value
\begin{equation}
{\cal A}_{\infty} = \ln \frac{\kappa_{+}\bar{c}_{\rm A}}{\kappa_{-}\bar{c}_{\rm B}} .
\label{A-inf}
\end{equation}

On the short time scale $t \ll D_k^{-1}R^2/(1+{\rm Da})^2$,
equations~(\ref{W+early}) and~(\ref{W-early}) with equation~(\ref{NESS}) show that the rates are given at early times by
\begin{eqnarray}
W_t^{(+)} &=& \frac{4\pi R^2\kappa_+}{\tilde\Delta} \left[\bar{c}_{\rm A}+\kappa_-R\left(1-\frac{R}{L}\right)\left(\frac{\bar{c}_{\rm A}}{D_{\rm B}}+\frac{\bar{c}_{\rm B}}{D_{\rm A}}\right)\right] + O(t) \, , \nonumber\\
W_t^{(-)} &=&  \frac{4\pi R^2\kappa_-}{\tilde\Delta} \left[\bar{c}_{\rm B}+\kappa_+R\left(1-\frac{R}{L}\right)\left(\frac{\bar{c}_{\rm A}}{D_{\rm B}}+\frac{\bar{c}_{\rm B}}{D_{\rm A}}\right)\right] + O(t)\, ,
\end{eqnarray}
and the corresponding affinity is
\begin{equation}
{\cal A}_t = \ln\frac{\kappa_+\bar{c}_{\rm A}+\kappa_+\kappa_-R\left(1-\frac{R}{L}\right)\left(\frac{\bar{c}_{\rm A}}{D_{\rm B}}+\frac{\bar{c}_{\rm B}}{D_{\rm A}}\right)+O(t)}{\kappa_-\bar{c}_{\rm B}+\kappa_+\kappa_-R\left(1-\frac{R}{L}\right)\left(\frac{\bar{c}_{\rm A}}{D_{\rm B}}+\frac{\bar{c}_{\rm B}}{D_{\rm A}}\right)+O(t)} \, ,
\end{equation}
which holds for small enough times.  Therefore, the affinity can take an early-time value that is much smaller than its asymptotic value~(\ref{A-inf}).

Taking equation~(\ref{Aff-inter-3D}) with $D=D_{\rm A}=D_{\rm B}$ and $\kappa=\kappa_{\pm}$, we see that the affinity reaches its asymptotic value if $t \gg D^{-1} R^2/(1+{\rm Da}^{-1})^2$.
This condition reads $t \gg D^{-1} R^2$ in the diffusion-limited regime where ${\rm Da}\gg 1$,
and $t \gg {\rm Da}^2 D^{-1} R^2$ in the reaction-limited regime where ${\rm Da}\ll 1$.
Since the molecular diffusivities typically take the value $D\simeq 10^{-9}$ m$^2$/s, the crossover time for a catalytic particle of micrometric radius is of the order of milliseconds.

\subsection{Numerical results}

For the simulation results, we suppose that the diffusion coefficients and the forward and reverse rate constants are equal, $D\equiv D_{\rm A}=D_{\rm B}$ and $\kappa\equiv \kappa_{\pm}$.  The function $\Psi(t)$ in equation~(\ref{Psi}) is given by
\begin{equation}
\Psi(t) = \frac{\ell^2 \kappa^2}{D^2}  \left( \bar{c}_{\rm A}+ \bar{c}_{\rm B}\right) \, \Upsilon(t)
\label{Psi-3D-equal}
\end{equation}
with
\begin{equation}
\Upsilon(t) = 4\pi \int_R^L dr \, u(r) \, \left[u(r)-v(r,t)\right] ,
\end{equation}
where $u(r) = \ell^{-1}\tilde\Delta^{-1} R^2 \left(1-r/L\right)$ and $\tilde\Delta \equiv 1 + 2\kappa D^{-1}R\left(1 - R/L\right)$.
The function $v(r,t)$ is the solution of the following problem:
\begin{eqnarray}
&&\partial_t v(r,t)= D\,\partial_r^2 v(r,t) \, , \\
&& R\, \partial_r v(R,t) = \left( 1 + \frac{2\kappa R}{D} \right) v(R,t) \, , \\
&& v(L,t) = 0 \, , \\
&& v(r,0)=u(r) \, ,
\end{eqnarray}
which can be solved numerically by spatial discretization into $I$ cells of size $\Delta r= (L-R)/I$.

\begin{figure}[h]
\centerline{\scalebox{0.5}{\includegraphics{fig2.eps}}}
\caption{Spherical catalytic particle: Stationary profile of the concentration difference $c_{\rm A}-c_{\rm B}$ versus the radial distance $r$ for the simulation with the parameter values~(\ref{3D-ex}). The dots are the simulation data and the solid line the theoretical result.}
\label{fig2}
\end{figure}

\begin{figure}[h]
\centerline{\scalebox{0.5}{\includegraphics{fig3.eps}}}
\caption{Spherical catalytic particle: Affinity ${\cal A}_t$ versus time $t$ for the simulation with the parameter values~(\ref{3D-ex}).  The dots are the simulation data and the solid line the theoretical result. The open squares show the affinity directly measured with $\ln[P(n,t)/P(-n,t)]$, while the filled squares show the affinity obtained from the Gaussian fit~(\ref{Gaussian_fit}).}
\label{fig3}
\end{figure}

These theoretical expressions are compared with numerical simulations based on a diffusive random walk model as well as multiparticle collision dynamics.

We first discuss the results obtained when the A and B particles move according to a diffusive random walk process between a spherical catalytic particle of radius $r=R$ and an outer sphere of larger radius $r=L$ where the molecules have the fixed concentrations $\bar{c}_{\rm A}$ and $\bar{c}_{\rm B}$. The simulation of the random walk process is described in Appendix~\ref{app:A}. The system contains a total of $N=522909$ A and B particles with $r<L$. The parameters take the following values:
\begin{equation}
D=1 \, , \qquad
\kappa = 0.4 \, , \qquad
R = 0.5 \, , \qquad
L=5 \, , \qquad
\bar{c}_{\rm A} = 526.1169 \, , \qquad
\bar{c}_{\rm B} = 473.5751 \, . \label{3D-ex}
\end{equation}
The process is at the crossover between the reaction- and diffusion-limited regimes because ${\rm Da}=2\kappa R/D=0.4$ and $\tilde\Delta =  1 + {\rm Da}\left(1-R/L\right) = 1.36$.  The mean reaction rate is ${\cal J} = 48.55$ and
the zero-time properties are given by
\begin{equation}
W_0^{(+)} = 652.40  \, , \qquad
W_0^{(-)} = 603.85  \, , \qquad
{\cal D}_0 = 628.13  \, , \qquad
{\cal A}_0 = 0.077329  \, ,
\end{equation}
while the asymptotic properties are
\begin{equation}
W_{\infty}^{(+)} = 486.13 \, , \qquad
W_{\infty}^{(-)} = 437.58 \, , \qquad
{\cal D}_{\infty} = 461.86 \, , \qquad
{\cal A}_{\infty} = 0.10521 \, .
\end{equation}
These theoretical values compare favorably with the computational results: ${\cal J} \simeq 49.7$ and ${\cal D}_{\infty} \simeq 465$. The stationary profile of the concentration difference $c_{\rm A}-c_{\rm B}$ is depicted in figure~\ref{fig2} and is also in agreement with the theoretical expectation~(\ref{phi}).

The finite-time affinity is shown in figure~\ref{fig3} where the computational results (dots) are compared with theory (solid line) obtained by spatial discretization into $I=1000$ cells.  The affinity is directly measured from simulation data as the slope of $\ln[P(n,t)/P(-n,t)]$ versus the number $n$ of reactive events during the time interval $[0,t]$ (open squares with error bars).  As time increases, the overlap between the probability distributions $P(n,t)$ and $P(-n,t)$ rapidly decreases.  In order to overcome this difficulty, the Gaussian probability distribution
\begin{equation}
P(n,t) \simeq \frac{1}{\sqrt{2\pi\sigma_t^2}} \, \exp\left[ -\frac{(n-\langle n\rangle_t)^2}{2\sigma_t^2} \right]
\label{Gaussian_fit}
\end{equation}
is fitted to the histogram. No significant deviations between the histogram and the Gaussian distribution have been observed. The affinity is thus estimated as ${\cal A}_t \simeq 2\, \langle n\rangle_t/\sigma_t^2$.
These values are depicted as filled squares in figure~\ref{fig3}, showing agreement between the simulation data and theory.

\vskip 0.3 cm

Microscopic simulations have also been carried out using a hybrid molecular dynamics-multiparticle collision dynamics scheme~\cite{MK99,MK00,K08,GIKW09}. In particular, we use the implementation with reversible catalytic reactions that satisfies detailed balance.~\cite{HSGK18} The catalytic particle resides in a system containing half inert solvent (S) particles and half reactive A and B particles. The roughly spherical catalytic particle is made of catalytic (C) beads connected by stiff harmonic springs. The fluid species interact with the particle beads through repulsive Lannard-Jones potential functions. Reversible reactions, ${\rm C}+{\rm A}\rightleftharpoons{\rm C}+{\rm B}$, take place on the catalytic beads with forward ($p_+$) and reverse ($p_-$) reaction probabilities. The reactive collision rule is designed such that the forward and reverse collisions satisfy the principle of detailed balance. To establish nonequilibrium conditions, the concentrations of A and B particles at a distance $L$ are controlled by relabelling particle species as A with probability $\bar{p}_{\rm A}$ or as B with probability $\bar{p}_{\rm B} = 1- \bar{p}_{\rm A}$ when a reactive particle moves across the boundary at $L$ from the region outside $L$ into the system. The resulting concentrations at $r = L$ are $\bar{c}_{\rm A} = \bar{p}_{\rm A}c_0 $ and $\bar{c}_{\rm B} = \bar{p}_{\rm B}c_0$, where $c_0 = \bar{c}_{\rm A}+\bar{c}_{\rm B}$ is the total concentration of A, B and S particles. This simulates a system where the concentrations outside of the sphere of radius $L$ are prescribed to be $\bar{c}_{\rm A}$ and $\bar{c}_{\rm B}$. The spherical particle is not fixed in space and undergoes Brownian motion. The sphere of radius $L$ is centered on the instantaneous position of the particle. See simulation details in Appendix~\ref{app:B}.

As above, we focus on the case where the diffusion coefficients and the forward and reverse rate constants are equal, $D\equiv D_{\rm A}=D_{\rm B}$ and $\kappa\equiv \kappa_{\pm}$. To determine $\kappa \equiv k^0/(4\pi R^2)$, we consider the irreversible reaction ${\rm C}+{\rm A}\to{\rm C}+{\rm B}$ on the particle with reaction probability $p_+ = 1$. The rate law for this case is $dc_{\rm A}(t)/dt = -k(t)n_{\rm C}c_{\rm A}(t)$, where $n_{\rm C} = 1/V$ is the Janus particle density. The time-dependent rate coefficient $k(t)$ starts at $k(0^+) = k^0$ and decays to the asymptotic value $k = k^0k_D/(k^0 +k_D)$ where $k_D = 4\pi DR$ is the diffusion-limited rate constant~\cite{TK04,HSGK18}. The time-dependent rate coefficient can be found by computing $k(t) = -[dc_{\rm A}(t)/dt]/[n_{\rm C}c_{\rm A}(t)]$ with the system starting from all A particles in the bulk without particle relabelling at the boundary $L$. The values obtained are $k^0\simeq 188.4 \pm 17.6$ and $k \simeq 3.68\pm 0.05$, from which one gets $k_D \simeq 3.75$. From $k_D$, one can estimate the outer edge of boundary layer to be at $R = k_D/(4\pi D)\simeq 5.0$, where $D = 0.0596$ is the fluid species diffusion coefficient. The Damk\"ohler number is equal to ${\rm Da} = 2\kappa R/D \simeq 100$, and $\tilde{\Delta}=80.2$, meaning that the system is in the diffusion-limited regime.

\begin{figure}[h]
\centerline{\scalebox{0.28}{\includegraphics{fig4.eps}}}
\caption{Spherical catalytic particle: Plots of $\langle n \rangle_t$, $\sigma^2_t$ and the finite-time affinity ${\cal A}_t$ versus time $t$ for two different reservoir conditions indicated in the figure inset. The dots are the simulation data and the solid lines the theoretical results.}
\label{fig4}
\end{figure}

The following two sets of parameter values with different reservoir concentrations have been used to obtain the results:
\begin{equation}
D=0.0596 \, , \qquad
\kappa = 0.6 \, , \qquad
R = 5.0 \, , \qquad
L=24 \, , \qquad
\bar{c}_{\rm A} = \{8,6\} \, , \qquad
\bar{c}_{\rm B} = \{2,4\} \, . \label{3D-ex-MPC}
\end{equation}
The mean reaction rates are ${\cal J} \simeq \{14.0, 4.7\}$, the zero-time properties are given by
\begin{equation}
W_0^{(+)} = \{941.7, 937.0\}  \, , \qquad
W_0^{(-)} = \{927.7, 932.4\}  \, , \qquad
{\cal D}_0 = \{934.7, 934.7\}  \, , \qquad
{\cal A}_0 = \{0.015, 0.005\}  \, ,
\end{equation}
while the asymptotic properties are
\begin{equation}
W_{\infty}^{(+)} = \{18.7, 14.0\}\, , \qquad
W_{\infty}^{(-)} = \{4.7, 9.3\}\, , \qquad
{\cal D}_{\infty} = \{11.7, 11.7\}\, , \qquad
{\cal A}_{\infty} = \{1.4, 0.4\}\, .
\end{equation}
These theoretical values compare favorably with the computational results: ${\cal J} \simeq
\{13.9,4.6\}$ and ${\cal D}_{\infty} \simeq \{13.6, 13.2\}$. The time-dependent $\langle n \rangle_t$, $\sigma_t^2$ and ${\cal{A}}_t$ obtained from theory and simulations are plotted in figure~\ref{fig4} where good agreement is seen.

Comparing figure~\ref{fig3} and figure~\ref{fig4}, we observe that the time-dependent affinity ${\cal A}_t$ starts from an early-time value ${\cal A}_0$ that is much closer to the asymptotic value ${\cal A}_{\infty}$ in the reaction-limited regime than in the diffusion-limited regime.  The reason is that the time-dependent affinity converges faster in the former regime than in the latter regime, as shown in subsection~\ref{S-geom-theory}.

\section{Janus catalytic particle}
\label{J-geom}

\subsection{Theory}

Here, we consider an immobile Janus particle of radius $R$ centered on the origin $r=0$ in the spherical coordinates $(r,\theta,\varphi)$.  The upper hemisphere is catalytic and the lower one is inert, so that the problem has a cylindrical symmetry under the rotations $\varphi\to\varphi+\alpha$ around the axis of the Janus particle.

The stationary problem~(\ref{phi1})-(\ref{phi4}) is given by
\begin{equation}
\nabla^2\phi =\frac{1}{r}\frac{\partial^2}{\partial r^2}\left(r\, \phi \right) +\frac{\hat{\cal L}}{r^2}\, \phi = 0 \, ,
\end{equation}
with the operator acting as $\hat{\cal L} Y_{lm}= -l(l+1) Y_{lm}$ on the spherical harmonics $Y_{lm}(\theta,\varphi)$.
The boundary conditions read
\begin{equation}
\left(\frac{\partial\phi}{\partial r}\right)_R = \frac{1}{\ell} (\phi-1)_R H(\cos\theta) \qquad\mbox{and}\qquad (\phi)_L=0 \, ,
\end{equation}
where $H(\xi)$ is Heaviside's function such that $H(\xi)=1$ if $\xi = \cos \theta >0$ and zero otherwise.  The solution of this problem can be expressed as
\begin{equation}
\phi(r,\theta) = {\rm Da} \sum_{l=0}^{\infty} a_l \, \left[ \left(\frac{R}{r}\right)^{l+1} - \left(\frac{R}{L}\right)^{l+1}\left(\frac{r}{L}\right)^l \right] \, P_l(\cos\theta)
\label{phi-Janus}
\end{equation}
in terms of the coefficients
\begin{equation}
a_l = \sum_{l'=0}^{\infty} \left({\boldsymbol{\mathsf M}}^{-1}\right)_{ll'} \, \int_0^1 d\xi \, P_{l'}(\xi)
\end{equation}
with
\begin{equation}
\left({\boldsymbol{\mathsf M}}\right)_{ll'} = \frac{2}{2l+1} \left[ l+1 + l \left(\frac{R}{L}\right)^{2l+1}\right] \delta_{ll'} + {\rm Da} \left[ 1 -\left(\frac{R}{L}\right)^{2l'+1}\right] \int_0^1 d\xi \, P_l(\xi) \, P_{l'}(\xi) \, ,
\label{Mll}
\end{equation}
where Da is the Damk\"ohler number~(\ref{Damkohler}).  The stationary mean concentrations are thus given by equation~(\ref{NESS}).

Here, the effective catalytic surface area~(\ref{Sigma}) is given by $\Sigma = 4\pi R^2 a_0 = 2\pi R^2 \left( 1 -{\rm Da} \, \gamma_J\right)$ with the constant
\begin{equation}
\gamma_J =\frac{1-2a_0}{\rm Da} = \sum_{l=0}^{\infty} a_l \left[ 1 - \left(\frac{R}{L}\right)^{2l+1}\right] \, \int_0^1 d\xi \, P_l(\xi) \, .
\label{gamma_J}
\end{equation}
In the limit $L\to\infty$, the effective catalytic surface area can be approximated by $\Sigma \simeq 2\pi R^2/(1+ 0.708115\, {\rm Da} )$, as shown in Ref.~\cite{GK18}.

The functions~(\ref{Upsilon}) can here be expressed as
\begin{equation}
\Upsilon_k(t) = 4\pi \int_R^L dr \sum_{l=0}^{\infty} \frac{1}{2l+1} \, v_{kl}(r,0) \left[ v_{kl}(r,0) - v_{kl}(r,t)\right]  ,
\label{Upsilon-Janus}
\end{equation}
by expanding the functions $f_k(r,\theta,t)$ as
\begin{equation}
f_k(r,\theta,t) = \frac{1}{r} \sum_{l=0}^{\infty} v_{kl}(r,t) \, P_l(\cos\theta)
\end{equation}
in terms of the solutions of
\begin{equation}
\partial_tv_{kl}(r,t) = D_k \left[ \partial_r^2 - \frac{l(l+1)}{r^2}\right] \, v_{kl}(r,t)
\label{Janus-vkl}
\end{equation}
with the boundary conditions
\begin{equation}
\left(\partial_r v_{kl}\right)_R =\frac{1}{R} (v_{kl})_R + \frac{2l+1}{2} \sum_{l'=0}^{\infty} \left(\frac{\kappa_+}{D_{\rm A}} \, v_{{\rm A}l'}+\frac{\kappa_-}{D_{\rm B}} \, v_{{\rm B}l'}\right)_R \int_0^1 d\xi \, P_l(\xi) \, P_{l'}(\xi)
\label{Janus-vkl-bc}
\end{equation}
and $(v_{kl})_L=0$, and starting from the initial conditions
\begin{equation}
v_{kl}(r,0) = {\rm Da} \, R \, a_l \, \left[ \left(\frac{R}{r}\right)^{l} - \left(\frac{R}{L}\right)^{l}\left(\frac{r}{L}\right)^{l+1} \right]
\label{Janus-vkl-0}
\end{equation}
for $k={\rm A}$ and $k={\rm B}$.

Therefore, the rates are given by equation~(\ref{W}) with
\begin{equation}
W_{\infty}^{(+)} = 2\pi R^2 \left( 1 -{\rm Da} \, \gamma_J\right) \, \kappa_+ \bar{c}_{\rm A} \, , \qquad\qquad
W_{\infty}^{(-)} = 2\pi R^2 \left( 1 -{\rm Da} \, \gamma_J\right) \, \kappa_- \bar{c}_{\rm B} \, ,
\label{W-inf-Janus}
\end{equation}
and the function (\ref{Psi}) is expressed in terms of the functions~(\ref{Upsilon-Janus}), which behave qualitatively as in the spherical geometry. If $L$ is finite, we have in the long-time limit that
\begin{equation}
\Upsilon_{\rm A}(\infty) = \Upsilon_{\rm B}(\infty) = \frac{4\pi}{3}\, a_0^2 \, \frac{R^4}{\ell^2} \, L \, \left[1+O(R/L)\right] \qquad\mbox{with}\qquad a_0=\frac{1}{2} \left( 1 -{\rm Da} \, \gamma_J\right) ,
\end{equation}
so that
\begin{equation}
\Psi(\infty) = \frac{4\pi}{3}\, a_0^2 R^4 L \kappa_+\kappa_- \left(\frac{\bar{c}_{\rm A}}{D_{\rm B}^2} + \frac{\bar{c}_{\rm B}}{D_{\rm A}^2} \right) \left[1+O(R/L)\right] ,
\label{Psi-Janus-t_inf}
\end{equation}
which is proportional to $L$ as in the spherical geometry.  Therefore, the rates~(\ref{W}) converge to their asymptotic value~(\ref{W-inf-Janus}) with corrections of $O(1/t)$, if $L$ remains finite.  However, if $L$ is infinite, the convergence proceeds with corrections of $O(1/\sqrt{t})$, as in the spherical geometry. Consequently, the affinity~(\ref{A-dfn}) converges towards its asymptotic value~(\ref{A-inf}) if $L$ is finite and infinite.

According to equation~(\ref{W-early}), the rates are given at early time by
\begin{eqnarray}
W_t^{(+)} &=& 2\pi R^2 \kappa_+  \left[ \bar{c}_{\rm A} - \gamma_J\, \frac{R}{D_{\rm A}} \left(\kappa_+\bar{c}_{\rm A} -  \kappa_-\bar{c}_{\rm B}\right)\right] + O(t) \, ,\\
W_t^{(-)} &=& 2\pi R^2 \kappa_-  \left[ \bar{c}_{\rm B} + \gamma_J\, \frac{R}{D_{\rm B}} \left(\kappa_+\bar{c}_{\rm A} -  \kappa_-\bar{c}_{\rm B}\right)\right] + O(t)\, ,
\end{eqnarray}
so that the early time behavior of the affinity is given by
\begin{equation}
{\cal A}_t = \ln\frac{\kappa_+  \left[ \bar{c}_{\rm A} - \gamma_J\, \frac{R}{D_{\rm A}} \left(\kappa_+\bar{c}_{\rm A} -  \kappa_-\bar{c}_{\rm B}\right)\right] + O(t)}{\kappa_-  \left[ \bar{c}_{\rm B} + \gamma_J\, \frac{R}{D_{\rm B}} \left(\kappa_+\bar{c}_{\rm A} -  \kappa_-\bar{c}_{\rm B}\right)\right] + O(t)} ,
\end{equation}
which can also be much smaller than the asymptotic affinity~(\ref{A-inf}).

\subsection{Numerical results}

We again suppose that the diffusion coefficients and the rate constants are equal, $D\equiv D_{\rm A}=D_{\rm B}$ and $\kappa\equiv \kappa_{\pm}$.  The finite-time affinity is thus given by equation~(\ref{A-dfn}) with the rates
\begin{eqnarray}
W_t^{(+)} &=& 2\pi R^2 \kappa \left(1-{\rm Da}\, \gamma_J\right) \bar{c}_{\rm A} + \ell^2\frac{\kappa^2}{D^2}\left(\bar{c}_{\rm A}+\bar{c}_{\rm B}\right) \frac{\Upsilon(t)}{t} \, ,\\
W_t^{(-)} &=& 2\pi R^2 \kappa \left(1-{\rm Da}\, \gamma_J\right) \bar{c}_{\rm B} + \ell^2\frac{\kappa^2}{D^2}\left(\bar{c}_{\rm A}+\bar{c}_{\rm B}\right) \frac{\Upsilon(t)}{t} \, ,
\end{eqnarray}
where the function $\Upsilon(t)=\Upsilon_{\rm A}(t)=\Upsilon_{\rm B}(t)$ is defined by equation~(\ref{Upsilon-Janus}) and calculated by solving equations~(\ref{Janus-vkl})-(\ref{Janus-vkl-bc}) for $v_l(r,t)\equiv v_{{\rm A}l}(r,t)=v_{{\rm B}l}(r,t)$.  This problem is solved numerically by spatial discretization $v_{l,i}(t) =v_l(r_i,t)$ with $r_i=R + (i-1/2)\Delta r$ where $i=1,2,...,I$ and $\Delta r = (L-R)/I$ with $I=20$ and $l\leq 100$ (see details in Appendix~\ref{app:C}).

\vskip 0.5 cm

Again, the theoretical results are compared with simulations where A and B particles move according to a diffusive random walk process between an immobile Janus particle of radius $r=R$ with a hemispherical catalytic surface and an outer sphere of larger radius $r=L$ where the molecules have the fixed concentrations $\bar{c}_{\rm A}$ and $\bar{c}_{\rm B}$, as described in Appendix~\ref{app:A}. The system contains a total of $N=523164$ A and B particles with $r<L$.  Here, the parameters have the following values:
\begin{equation}
D=1 \, , \qquad
\kappa = 0.4 \, , \qquad
R = 0.5 \, , \qquad
L=5 \, , \qquad
\bar{c}_{\rm A} = 527.195 \, , \qquad
\bar{c}_{\rm B} = 474.545 \, . \label{Janus-ex}
\end{equation}
The process is at the crossover between the reaction- and diffusion-limited regimes since ${\rm Da} =  2\kappa R/D = 0.4$. For this system, we have that $\gamma_J = 0.51960$. The mean reaction rate is given by ${\cal J} = 26.2 $, the time-zero properties by
\begin{equation}
W_0^{(+)} = 327.81  \, , \qquad
W_0^{(-)} = 301.60  \, , \qquad
{\cal D}_0= 314.71  \, , \qquad
{\cal A}_0 = 0.083318  \, ,
\end{equation}
and the asymptotic properties by
\begin{equation}
W_{\infty}^{(+)} = 262.40 \, , \qquad
W_{\infty}^{(-)} = 236.19 \, , \qquad
{\cal D}_{\infty} = 249.30 \, , \qquad
{\cal A}_{\infty} = 0.10521 \, .
\end{equation}
Here also, these theoretical values compare favorably with the computational results: ${\cal J} \simeq 26.5$ and ${\cal D}_{\infty} \simeq 250$. We notice that the diffusivity does not deviate much from its asymptotic value in the present case.

\begin{figure}[h]
\centerline{\scalebox{0.5}{\includegraphics{fig5.eps}}}
\caption{Janus catalytic particle: Affinity ${\cal A}_t$ versus time $t$ for the simulation with the parameter values~(\ref{Janus-ex}).  The dots are the simulation data and the solid line the theoretical result. The open squares show the affinity directly measured with $\ln[P(n,t)/P(-n,t)]$, while the filled squares show the affinity obtained from the Gaussian fit~(\ref{Gaussian_fit}).}
\label{fig5}
\end{figure}

The finite-time affinity is shown in figure~\ref{fig5} where the computational results (squares) are compared with theory (solid line).  As for the spherical catalytic particle, the affinity is directly measured from simulation data as the slope of $\ln[P(n,t)/P(-n,t)]$ versus the number $n$ of reactive events during the time interval $[0,t]$ (open squares with error bars), as well as by Gaussian fits to the histograms of $n$ values.  No significant deviations between the histogram and the Gaussian distribution have been observed.  The affinity is then estimated as ${\cal A}_t \simeq 2\, \langle n\rangle_t/\sigma_t^2$.  These values are depicted as filled squares in figure~\ref{fig5}, showing here also agreement between the simulation data and theory.

\begin{figure}[h]
\centerline{\scalebox{0.28}{\includegraphics{fig6.eps}}}
\caption{Janus catalytic particle: The mean number of reactive events $\langle n\rangle_t$, the corresponding variance $\sigma^2_t$, and the affinity~${\cal A}_t$ versus time $t$ for the microscopic simulation with the multiparticle collision method.  The dots are the simulation data and the solid lines the theoretical results.  The results are the average of $40$ realizations and the error is given by the standard deviation.}
\label{fig6}
\end{figure}

Microscopic simulations as described in Appendix~\ref{app:B} have also been carried out for a Janus particle made from catalytic and noncatalytic beads connected by stiff harmonic springs. We have chosen the interaction potentials of the A and B particles with the beads of the Janus particle to be equal so that the diffusiophoretic mechanism that leads to self propulsion does not operate. The Janus particle does, however, execute Brownian motion as is the case for the spherical particle. Again, we investigate the time-dependent reaction rates, diffusivities and affinity for two parameter sets with different reservoir concentrations:
\begin{equation}
D=0.0596 \, , \qquad
\kappa = 0.6 \, , \qquad
R = 5.0 \, , \qquad
L=24 \, , \qquad
\bar{c}_{\rm A} = \{8,6\} \, , \qquad
\bar{c}_{\rm B} = \{2,4\} \, . \label{3D-Janus-MPC}
\end{equation}
Here, the process evolves in the diffusion-limited regime because the Damk\"ohler number has the value ${\rm Da}=2\kappa R/D\simeq 100$.
The mean reaction rates are ${\cal J} \simeq \{9.7,3.23\}$, the zero-time properties are given by
\begin{equation}
W_0^{(+)} = \{475.8, 472.6\}  \, , \qquad
W_0^{(-)} = \{466.1, 469.4\}  \, , \qquad
{\cal D}_0 = \{471.0, 471.0\}  \, , \qquad
{\cal A}_0 = \{0.021,0.0068\}  \, ,
\end{equation}
while the asymptotic properties are
\begin{equation}
W_{\infty}^{(+)} = \{12.9, 9.7\}\, , \qquad
W_{\infty}^{(-)} = \{3.23, 6.47\}\, , \qquad
{\cal D}_{\infty} = \{8.09, 8.09\}\, , \qquad
{\cal A}_{\infty} = \{1.38,0.405\}\, .
\end{equation}
These theoretical values compare favorably with the simulation results: ${\cal J} \simeq
\{10.1,3.35\}$ and ${\cal D}_{\infty} \simeq \{9.3,10.6\}$. The time-dependent $\langle n \rangle_t$, $\sigma_t^2$ and ${\cal{A}}_t$ obtained from theory and simulations are plotted in figure~\ref{fig6} where good agreement is seen.

Here also, the comparison between figure~\ref{fig5} and figure~\ref{fig6} shows that the time-dependent affinity ${\cal A}_t$ remains closer to its asymptotic value ${\cal A}_{\infty}$ in the reaction-limited regime than in the diffusion-limited regime, confirming the general behavior already observed in the spherical geometry.

\section{Conclusion and perspectives}
\label{conclusion}

In this paper, the finite-time fluctuation theorem of Ref.~\cite{TDFT} has been investigated for the diffusion-influenced surface reaction ${\rm A}\rightleftharpoons{\rm B}$ on spherical and Janus catalytic particles.  The finite-time rates of the forward and reverse reactions ${\rm A}\rightleftharpoons{\rm B}$ have been analytically calculated in both geometries by solving diffusion equations with the special boundary conditions obtained in Ref.~\cite{TDFT}.  These rates provide the time-dependent thermodynamic force or affinity driving the process away from equilibrium.  This affinity converges towards its asymptotic value determined by the concentrations of the reacting species at the reservoir, as predicted by infinite-time fluctuation theorems.

The results show that the affinity may take a much lower value at early time than its expected asymptotic value.  The reason is that the affinity reaches its asymptotic value beyond the diffusion time characteristic of the reaction taking place on the catalytic particle.   In the reaction-limited regime, this diffusive time is short, so that the affinity rapidly converges towards its asymptotic value.  However, the diffusion time may be significantly longer in the diffusion-limited regime.  For micrometric catalytic particles and small diffusing molecules, the crossover time is of the order of miliseconds, which is short relative to macroscopic measurement times, hence justifying the use of the long-time limit.

Theoretical results for the spherical catalytic and Janus particles were compared with numerical simulations using two different methods: a random walk algorithm and an algorithm based on multiparticle collision dynamics. These systems were studied for diffusion-limited catalytic reactions as well as for reactions that lie in crossover regime between reaction- and diffusion-limited kinetics. In the diffusion-limited regime the affinity takes an early-time value that is significantly smaller than its asymptotic value. In all cases theoretical and simulation results are in agreement.

In the three-dimensional geometries of the spherical and Janus catalytic particles, the crossover time does not depend on the distance between the catalytic surface and the reservoir where the concentrations of reacting species are fixed.  This is no longer the case in the one-dimensional planar geometry, as we shall report in a future publication.

\section*{Acknowledgments}

Financial support from the International Solvay Institutes for Physics and Chemistry, the Universit\'e libre de Bruxelles (ULB), the Fonds de la Recherche Scientifique~-~FNRS under the Grant PDR~T.0094.16 for the project ``SYMSTATPHYS", the Belgian Federal Government under the Interuniversity Attraction Pole project P7/18 ``DYGEST", and the Natural Sciences and Engineering Research Council of Canada is acknowledged.

redpajama_set_name: RedPajamaArXiv


A guide to Microlighting over Angkor Wat, Cambodia.
Angkor Wat in Cambodia has more than a thousand temples across an area that stretches 400 square kilometres. The temples come in various states of disrepair, shapes, sizes and magnificence.
Most famous because of how old they are (built from 9th-12th century) and because of the astonishing Cambodian scenery as a backdrop.
Also known for being in the film Lara Croft: Tomb Raider from 2001.
Best Ways to Explore the Temples?
Although I am recommending Microlighting over Angkor Wat this should always be in conjunction with seeing at least the main temples by foot for a number of reasons.
Temple Up-keep: I mentioned earlier that the temples are in various states of disrepair and many of the temples are well looked after so tourists like you and me can see them in all their glory. Going by foot means you will have to pay an entrance fee to see the temples and this goes towards their up-keep (roughly 25% of your fee).
Entrance Pass Costs: $20 for a 1 day pass, $40 for a 3 day pass and $60 for a 7 day pass.
Another Perspective: Of course seeing something from the air is completely different from seeing it from the ground, both are spectacular but in completely different ways. There's more culture and a chance to meet locals when your on foot as well.
Explore the Detail Engravings: From the air you can't see any of this but from the ground you realise that's these magnificent structures are not just plan block structures but you'll see the faces at temple Angkor Thom or the engravings at Angkor Wat.
So you can Pretend you're Tomb Raider: Ta Prohm Temple is the temple from the Lara Croft: Tomb Raider film from 2001. Well known for its trees growing through the temple structure (although many other temples have this as well).
When I went out by foot we took a Tuk Tuk taxi to Angkor Wat for Sunrise then did a mixture of walking and taxi's to the other temples.
Well this is easy as far as I can see, there is only one company you can go with in Siem Reap, although there are other companies that offer plane or helicopter trips. Both this other options cost a considerable amount more and can't fly as low as a Microlight can, plus you're not out in the fresh sky air either.
The company I went with was called Sky Venture, unfortunately the British owner had to return to the UK after a motorbike accident but luckily two year's later the pilot that I flew with from this company is back and flying the same tour I went but through a different firm.
Prices range from US $75-$225 depending on what tour you want to do. I suggest going for the midrange tour costing $125, which allows you to see Angkor Wat as I did plus surrounding farmlands and the Roluos Group Temples.
There open everyday from 7am to 11am and 3:30pm until dark.


text:
https://celticswire.usatoday.com/lists/nba-boston-celtics-williams-y-white-on-winning-game-6/
'This isn't a team that's just going to lie down,' says Boston's Grant Williams ahead of a potential Heat closeout tilt in Game 6
(AP Photo/Michael Dwyer)
May 27, 2022 1:06 pm ET
The Boston Celtics could punch their ticket to the 2022 NBA Finals with a win against the Miami Heat in Game 6 of their Eastern Conference finals with the Jimmy Butler-led squad on Friday. To do it, they will have to defeat a team full of proud veterans who will not go quietly into the night.
Aware that the series is not over and that, even in their currently hobbled state, the Heat can still rip a win out of the hands of a team that does not respect them, Boston's Derrick White and Grant Williams both had words about the job ahead after practice on Friday.
"We don't want to give them life," explained White.
Miami Heat at Boston Celtics: 2022 NBA playoffs Game 6 Eastern Conference finals (5/27) https://t.co/Kl5V6zzbP8
— The Celtics Wire (@TheCelticsWire) May 27, 2022
"Closeout games are the toughest games," he explained. "We know they are going to come out and up their physicality and do things that they need to do to stay in this series."
"You can't relax and think it's over — they are not going to give up, we have got to be ready to go."
Dunkin' Donuts and Puma are teaming for a new sneaker ahead of this year's Iced Coffee Day. Marcus Smart shared his excitement on his Instagram story. https://t.co/ksFRXqaKAJ
In response to comments already placing the Celtics in the finals with the Golden State Warriors after the Dubs closed out the Dallas Mavericks on Thursday night, White had some words as well.
"It's just the nature of the game. You lose one game, they think you're done. You win one game, they think you are going to win it all. So, we're just trying to stay consistent, not get too high and not get too low, to not really listen to that. That stuff doesn't matter."
"Miami is a great team," he added. "We can't look past them."
Celtics Lab 115: Diving into whether Boston can close out their East Finals series vs. the Miami Heat in Game 6 with Gary Gulman https://t.co/R3g6LMLSgC
Williams had a similar perspective on the Heat. "You can't look past the team at any stage," he suggested. "You can't think about the next game before you take care of the one that you're focused on."
"For this team, especially, this is a team that's going to come out and give you their best shot and their best punch. And you have to be willing to withstand that and also compete through it because this isn't a team that's just going to lie down. They don't have guys like that on their team. So for us, it's a matter of understanding that today is the most important day for us; you can't look past it. You want to take care of this on your home court and you want to make sure that now you come out with the right intensity but you come out engaged because you don't want to give any life to a team that may be dealing with injuries and everything like that."
'You don't want to give them any chance to have that exuberant step to come back and hopefully bounce back," Williams said.
One intrepid analyst tried their hand at naming them. https://t.co/JdeFev1Uld
"So for us, we were in that position last area. So we understand the intensity and physicality and everything that you have to play with and expect to see.
So for us, it's just a matter of how we're going from … the hunter to the hunted. So, we have to do a great job of adapting to that and being consistent in our own mentality."
It is his second time making an All-NBA team, and his best showing yet doing so. https://t.co/Ow9hDPMep3
Boston's Jayson Tatum elected to 2021-22 All-NBA first team
Celtics Lab 114: Are there any adjustments left? Exploring the Boston-Miami East finals with Kyle Russell
Marcus Smart teases new Dunkin' Donuts inspired PUMA shoes
'If I was going to fail, I was going to do it aggressively,' said Derrick White of his mindset going into Game 4 vs. Heat
Three keys to a Boston Celtics trip to the 2022 NBA Finals
Heat at Celtics: Boston beats up on the Heat at the Garden 102-82 to win Game 4
'I've been a chatterbox my whole life,' says Boston's Grant Williams of the personality quirk that drives his elite play


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News Nicaragua Segments
International Federation for HR Condemns Ortega's Repression in Nicaragua
October 27, 2019 Circles Robinson 0 Comments
They demand an immediate end to the persecution of human rights organizations by the Ortega-Murillo regime.
By Yader Luna (Confidencial)
HAVANA TIMES – Non-governmental human rights organizations from 134 countries that make up the International Federation for Human Rights (FIDH) condemned the brutal repression and violation of human rights towards the Nicaraguan people by the Ortega-Murillo regime.
Gathered during their 40th Congress in Taipei, Taiwan, they unanimously approved the resolution on Nicaragua which "demands the Nicaraguan State to immediately cease persecution of human rights organizations."
They also call for the devolution of the Nicaraguan Center for Human Rights (Cenidh) legal status as well as those of another eight non-governmental organizations which were shut down by the dictatorship.
The human rights organizations request the Office of the United Nations High Commissioner for Human Rights and the Inter American Commission on Human Rights (IACHR) to "maintain their watch on the Ortega Murillo regime and monitor the persistent repression it has unleashed in Nicaragua."
The resolution, presented by Cenidh and approved by the other organizations, calls on the international community to "maintain and increase the pressure upon the regime in Nicaragua and to take the necessary measures that would allow the serious and persistent violations of human rights to stop, and judge and punish those responsible." Likewise, the groups call for "individual sanctions against the perpetrators in the same way some governments have been doing."
Follow up on the violations
The International Federation for Human Rights points out that since April 2018, when the protests against the dictatorship started, "the human rights crisis worsened which resulted in at least 322 assassinations left in impunity, more than 70 thousand in exile, hundreds wounded, political prisoners, tortures and abductions, as well as the persistent violation of public freedoms and the establishment of a State of terror characterized by the excessive use of police and paramilitary forces".
"As part of their authoritarian policy, Nicaraguan President Daniel Ortega and his wife, Vice-President Rosario Murillo's regime expelled the Inter American Commission on Human rights (IACHR) and the Office of United Nations High Commissioner for Human Rights (OHCHR) missions from the country, who were following up on the country's severe social political and human rights crisis, as well as the Interdisciplinary Group of Independent Experts (GIEI), which confirmed and held the State responsible for the execution of violations classified as crimes against humanity", they explained.
The resolution indicates that "the pattern of governmental violence includes persecution, hounding, harassment and stigmatization of human rights defenders whose right to defend rights has been restricted".
"One of the aggressions' targets has been the organization affiliated with the FIDH, the Nicaraguan Center for Human Rights (Cenidh), arbitrarily stripped of its legal status, its assets seized and its members persecuted", adds the resolution.
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In September, at the same conference I'm at right now (CEO Space), someone asked me, "If you had a billion dollars, what would you wake up and do every day?" Even though I love my event business, the answer was easy - "I'd do music, write music, record & perform music, go to concerts, watch music online, etc... I'd immerse myself in music." Long story short, he gently nudged me in that direction.
Then I started sharing the conversation with other folks I met here (at CEO Space) because I started to feel torn about how to run my company while realizing that my passions were elsewhere. One of them encouraged me to just move "4%" more in the direction of my dreams every day. 4% seems inconsequential but it makes a huge difference over time and it's very manageable - no excuses!
He also introduced me to Vinny Ribas who teaches artists how to have an independent music career! 4% each day has eventually culminated in me working with songwriters that Vinny has connected me to, writing my own songs with Vaughn Fahie Jazz and recording an album that's set to release in August.
So, I have two questions for you!
- What would you really do if you had a billion dollars (if money was no object)?
- If you're not yet doing it, what would it look like to move 4% more in the direction of your dreams each day?
Start TODAY! It's just 4%!!! You can do it!!!


I was excited to see all of the summer street projects. I'm sure they will be great when completed ("Paving for Progress: See what roads are on the list for Cedar Rapids," March 15).
1. Visitors to our town may not know "an alternate route" when they suddenly come upon a "Road Closed" sign. Reasonable detour directives are appreciated.
2. Those who do live here and know one or two alternate routes would appreciate at least one of those remaining open.
3. Don't start a job if you don't have the manpower to complete it in a timely fashion. It would be my guess that the city would not be satisfied if they were on the receiving end of a stalled construction project that resulted in prolonged inconvenience and disruption.
Any chance of fixing the bottom of Green Valley Terrace while doing Broadlawn? Thanks and I look forward to the improvements.


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Listen to your music, playlists or radio anywhere and everywhere in your home.
Lighting in a Smart Home works best when automated, or with simple rules applied.
Free your TV from the living room.
Secure your home, keep your loved ones safe.
You know that you want a Smart Home, but you're not sure where to start? Get in touch – we'll help you understand what will be the best way to achieve the home of your dreams.
"Staff are very friendly, knowledgeable and willing to discuss technical options"


Leaving Budapest for the summer, the buildings mock me. They stand resplendent in their morning veil as I glance back over my shoulder Buda-wards, not fearing to become a pillar of salt, merely to find hidden in the secrets of those spires and ornately tiled roofs, a sentiment, a love for this city.
Pest greets me as I cycle through it, the hustle and bustle, the noise, the traffic, the bars.
I know it's early but whoever came up with that silly notion of it being too early to have a beer! The postman crosses swords with the all-night reveller in a kocsma, both on dawn patrol.
Soon I'll be home, to Ireland, my first home, but having made this a clear second, the other places in which I've resided shuffle for attention. They'll get it in moments of reminiscence, but for now I must contend with the idea that for two months, while alleviating homesickness, I may also become homesick.


Brazilian Jiu-Jitsu (BJJ) is a martial art that revolves around grappling. The art is designed to teach an individual to control a resisting opponent in ways that force him to submit. Jiu-Jitsu focuses on taking an opponent down to the ground and wrestling for a dominant control position.
Brazilian Jiu-Jitsu is a sport in which a person with lesser size, strength, and aggression can control and overcome an individual with greater size, strength, and aggression. To do this, students learn to utilize superior leverage, grip, and position on their opponent. Students that practice the sport will gain a broader understanding of the mechanism and limits of the human body.
The art of Jiu-Jitsu is both physically and mentally demanding allowing practitioners to utilize problem solving and critical thinking skills. Aside from the self-defense benefits, students increase physical fitness, increase mental focus, and gain self-knowledge of their body. Most importantly, students are able to learn and grow with people that share the same passion.
Classes at the academy are taught four days a week and start with a strength building warm up and are followed by demonstration and practice of different techniques. Students also have the opportunity to participate in live grappling with other students that are on or near the same level. Overall, Brazilian Jiu-Jitsu is a great workout and teaches realistic self-defense application that both men and women can utilize and enjoy.
Whether you are looking to compete OR train solely for self-defense purposes, Renzo Gracie Ozone Park will cater to your needs. Our highly skilled black belt professor is dedicated to coaching and guiding our students. We have proudly competed in numerous tournaments such as Good Fight, North American Grappling Association (NAGA), New Breed, Pride, and much more! Our students have placed gold in multiple events effectively displaying their skills. For those looking to train solely for self-defense purposes, Renzo Gracie Ozone Park provides students with the knowledge to defend themselves confidently.
Why Choose Renzo Gracie Ozone Park?
Renzo Gracie Ozone Park/Training Zone NYC Inc. is the only school in the area that is affiliated with the Gracie system of teaching. Renzo Gracie is a world-renowned practitioner of Brazilian Jiu-Jitsu and is the grandson of the late Carlos Gracie, founder of Brazilian Jiu-Jitsu. The Gracie family consists of Jiu-Jitsu professors and fighters who have changed the current state of martial arts instruction worldwide. This system of teaching has proven effective with numerous professional fighters such as Georges St-Pierre, Frankie Edgar, Chris Weidman, and Matt Serra. By choosing our academy, you are learning Jiu-Jitsu from a Renzo Gracie and John Danaher black belt professor with over 13 years of experience.