text:
\section{Introduction}

In a thermodynamic model, Di\'osi {\it et al.} \cite{Diosi} considered an elementary model of a reservoir,
in which the thermodynamic entropy production due to the application of an
external field equals the change of
the informatic entropy. Let us recapitulate the main ideas of this model. It has been assumed that this reservoir is composed by
$n$ distinguishable particles with finite energy levels. The time evolution of these particles is governed by the
non-interacting Hamilton operator
\begin{equation}
 H_R=H\otimes I^{\otimes (n-1)}+I \otimes H \otimes I^{\otimes (n-2)}+ \dots +
 I^{\otimes (n-1)} \otimes H. \nonumber
\end{equation}
Each particle is considered to be in a Gibbs state with a fixed inverse temperature $\beta$ and the state of the reservoir
is assumed to be completely uncorrelated:
\begin{equation}
 \rho_R=\rho^{\otimes n}, \quad \rho=\frac{\ee^{-\beta H}}{\mathrm{Tr} \{ \ee^{-\beta H} \}}.
 \label{Gibbs}
\end{equation}
An external field is applied instantaneously on the reservoir, i.e a unitary
evolution takes place on a much smaller time scale than the time
scale of the reservoir's evolution, and it is assumed that only one of the particles changes its state to $\sigma$:
\begin{equation}
 \sigma=U \rho U^*,\quad [U, H] \neq 0, \nonumber
\end{equation}
where $U$ is a unitary operator. This results in $n$ possible reservoir states $\rho'_R$
\begin{equation}
 \sigma \otimes \rho^{\otimes (n-1)},\, \rho \otimes \sigma \otimes \rho^{\otimes (n-2)},\, \dots,\, \rho^{\otimes (n-1)} \otimes \sigma, \nonumber
\end{equation}
depending which particle changed its state. The reservoir changes its mean energy by
\begin{equation}
 \Delta E = \mathrm{Tr}\{H_R \rho'_R\}-\mathrm{Tr}\{H_R \rho_R\}=\beta^{-1} S(\sigma || \rho)>0 \nonumber
\end{equation}
where $S(\sigma || \rho)$ is the relative entropy and we used the relation
$\log \rho=-\beta H- \log (\mathrm{Tr} \{ \ee^{-\beta H} \})$(see
Eq. \eqref{Gibbs}) together with the
fact that the von Neumann entropy is unitarily invariant:
\begin{equation}
 S(\sigma)=-\mathrm{Tr}\{\sigma \log (\sigma)\}=-\mathrm{Tr}\{U\rho U^* \log (U\rho U^*)\}=S(\rho). \nonumber
\end{equation}
If this mean energy
is dissipated in the reservoir, then the thermodynamic entropy production is
equal to $S(\sigma || \rho)$. But
the informatic entropy production is zero:
\begin{equation}
  S(\rho'_R)- S(\rho_R)=0. \nonumber
\end{equation}
The authors of Ref.~\cite{Diosi}
needed a new state $R_n$ instead of $\rho'_R$ such that
\begin{equation}
 \mathrm{Tr}\{H_R \rho'_R\}=\mathrm{Tr}\{H_R R_n\} \quad \mathrm{and} \quad S(R_n)-S(\rho'_R) \rightarrow S(\sigma||\rho) \nonumber
\end{equation}
in the thermodynamic limit $n \to \infty$. It was conjectured that for
\begin{equation}
R_n=\frac{\sigma \otimes \rho^{\otimes (n-1)}+\rho \otimes \sigma \otimes \rho^{\otimes (n-2)}+\dots+ \rho^{\otimes (n-1)} \otimes \sigma}{n}
\nonumber
\end{equation}
the informatic entropy production rate is equal to the relative entropy $S(\sigma||\rho)$ in the thermodynamic limit. This conjecture was proved
by Csisz\'ar {\it et al.} \cite{Petz}. The proof generalizes the conjecture of Ref.~\cite{Diosi} for arbitrary density matrices $\sigma$ and $\rho$,
which are not related any more by a unitary transformation, i.e. possibly
$S(\rho'_R)\neq S(\rho_R)$, and shows that
\begin{eqnarray}
 &&\lim_{n \to \infty} \Big(S(R_n)-S(\rho'_R)\Big)= S(\sigma||\rho), \label{entconv} \\
 &&\forall \rho'_R \in \{\sigma \otimes \rho^{\otimes (n-1)},\, \rho \otimes \sigma \otimes \rho^{\otimes (n-2)},\, \dots,\,
 \rho^{\otimes (n-1)} \otimes \sigma\}. \nonumber
\end{eqnarray}

This whole problem can be viewed from a different angle. Let us consider the approach of M. J. Donald \cite{Donald} and calculate the Helmholtz
free energy $F$ of the reservoir in the state $R_n$ and $\rho_R=\rho^{\otimes n}$. The difference between them is given by
\begin{equation}
 F(R_n)-F(\rho^{\otimes n})=\beta^{-1} S(R_n||\rho^{\otimes n}). \nonumber
\end{equation}
One can simply compute
\begin{eqnarray}
 S(R_n||\rho^{\otimes n})=-S(R_n)+S(\rho'_R)+S(\sigma||\rho). \nonumber
\end{eqnarray}
Thus Eq. \eqref{entconv} is equivalent to
\begin{equation}
 S(R_n||\rho^{\otimes n}) \rightarrow 0 \quad \mathrm{as} \quad n \to \infty, \nonumber
\end{equation}
which yields
\begin{equation}
 F(R_n)-F(\rho^{\otimes n}) \rightarrow 0 \quad \mathrm{as} \quad n \to \infty. \nonumber
\end{equation}

This means that in the thermodynamic limit the reservoir changed its equilibrium state to another equilibrium state. A natural question is to find
all these thermodynamically equivalent states. In
case
when only one particle changed its Gibbs state $\rho$ to an arbitrary
state $\sigma$, the proof in Ref.~\cite{Petz} shows that the state $R_n$
-- where the randomization is uniform, i.e. each individual state has the same
probability to change to state $\sigma$ -- is
an equilibrium state equivalent to the inital state $\rho^{\otimes n}$.

In this paper we consider a general case:
We suppose that the randomization is not uniform, hence
we consider the state
\[
 \rho_n = a_{n,1} \sigma \otimes \rho^{\otimes (n-1)}+ a_{n,2} \rho \otimes
 \sigma \otimes \rho^{\otimes (n-2)} +\dots
 + a_{n,n} \rho^{\otimes (n-1)}
 \otimes \sigma, \label{rhon}
\]
where the matrix $(a_{i,j})$ contains the changing weights for all $n$, such that $a_{n,i} \in [0,1]$ when $i \leqslant n$ otherwise $a_{n,i}=0$
and $\sum^n_{i=1}a_{n,i}=1$. This construction allows us to study the
thermodynamic limit $n \to \infty$. The main question is about the properties of this matrix
that assure the following limit:
\begin{equation}
S(\rho_n||\rho^{\otimes n}) \rightarrow 0 \quad \mathrm{as}  \quad n \to \infty. \nonumber
\end{equation}

In general, a matrix $(a_{i,j})$ is regular or T-matrix (see Ref.~\cite{Cooke}), if the following conditions are satisfied:
\begin{eqnarray}
 &&\sup_{n \in \mathbb{N}^+} \sum^{n}_{i=1} |a_{n,i}|< \infty, \quad \lim_{n \to \infty} \sum^{n}_{i=1} a_{n,i}=1, \nonumber \\
 &&\lim_{n \to \infty} a_{n,j}=0,\quad \forall j\in \mathbb{N}^+. \nonumber
\end{eqnarray}
It is strongly regular if it is regular and (see Ref.~\cite{Jardas})
\begin{equation}
 \lim_{n \to \infty} \sum^\infty_{j=1} |a_{n,j+1}-a_{n,j}|=0. \nonumber
\end{equation}

We are going to give an analytical proof of the  following statement:
\begin{thm}\label{thm1}
 Let $\rho$ be a Gibbs state and $\sigma$ a state such that $\mathrm{supp}(\sigma) \subseteq \mathrm{supp}(\rho)$. Then for every
 strongly regular matrix $(a_{i,j})$, where $a_{i,j}\ge 0$ for all
 $i,j\in\mathbb{N}^+$ and $\sum_{i=1}^n a_{n,i}=1$ for all $n\in \mathbb{N}^+$,
 and
\[
 \rho_n = a_{n,1} \sigma \otimes \rho^{\otimes (n-1)}+ a_{n,2} \rho \otimes
 \sigma \otimes \rho^{\otimes (n-2)} + \dots
 + a_{n,n} \rho^{\otimes (n-1)}
 \otimes \sigma
\]
we have
\begin{equation}
 \lim_{n \to \infty} S(\rho_n||\rho^{\otimes n})=0. \nonumber
\end{equation}
\end{thm}

The entries of the strongly regular matrix $(a_{i,j})$ must be restricted to the
interval $[0,1]$ in order for the state in \eqref{rhon} to be a
real physical state of the reservoir. Thus, Theorem \ref{thm1} is capable to identify a large class of states for which the Helmholtz
free energy in the thermodynamic limit is the same as for the equilibrium state $\rho^{\otimes n}$, where $n \in \mathbb{N}^+$. The presented
proof combines the idea that comes from Ref.~\cite{Petz} with results from ergodic theory \cite{Krengel}.

\section{GNS approach}

Let us start with Eq. \eqref{rhon}: Applying the same idea that was used in
Ref.~\cite{Petz} we can rewrite $\rho_n$ in a more suitable form. Let $X=\rho^{-1/2}\sigma\rho^{-1/2}$, where $\rho$ is invertible
(see definition in \eqref{Gibbs}). Then we have
\[
\rho_n = \left(\rho^{\frac{1}{2}}\right)^{\otimes n} \Big( a_{n,1} X \otimes
I^{\otimes (n-1)} + a_{n,2} I \otimes X \otimes I^{\otimes (n-2)}
+\cdots + a_{n,n} I^{\otimes (n-1)} \otimes X\Big)
\left(\rho^{\frac{1}{2}}\right)^{\otimes n}.
\]

Here, we cannot use directly the mean ergodic theorem that was applied in
Ref.~\cite{Petz} or Ref.~\cite{Ohya} (see Proposition 1.17 there).
Nevertheless, we will follow a very similar path to that which was given in the
proof of the aforementioned proposition. The algebra underlying our problem is the same as the proposition's
and hence we are going to adopt the notation used in Ref.~\cite{Ohya}. As we have
already stated, the reservoir consists of
particles with finite energy levels, and we denote by $d$ the dimension of the
Hilbert space underlying their quantum mechanical
description. Let $M_d$ denote the set of all $d \times d$ complex
matrices acting on this Hilbert space. Since $M_d$ is a $C^*$-algebra
with identity, we consider the family of $C^*$-algebras
$\{M_{d^n}=(M_d)^{\otimes n}| n \in \mathbb{N}^+\}$ the $n$-fold algebraic
tensor products of $M_d$. If we identify $M_{d^n}$ with $M_{d^n}\otimes I$ in
$M_{d^{n+1}}$, where $I=I_d$ denotes the identity operator on the $d$ dimensional
Hilbert space, we can consider
$M_{\infty}=\bigcup_{n=1}^\infty M_{d^n}$, the inductive limit $\ast$-algebra of $\{M_{d^n}\}$, and its completion
$A_{\infty}$. In other words,
\[
A_{\infty}=\overline{\bigotimes_{i=1}^\infty A_i},
\]
where $A_i=M_d$ for all $i\in\mathbb{N}^+$
(for the details see Ref.~\cite{Sakai}, Section 1.23). The family of faithful states
(see Eq. \eqref{Gibbs}) $\{\varphi_n(x)=\mathrm{Tr} \{\rho^ {\otimes n} x\}| x
\in M_{d^n}, \, n \in \mathbb{N}^+\}$, are functionals on
the particular $C^*$-algebras, which extend to a faithful state
$\varphi_{\infty}$ on $A_{\infty}$. The GNS representation
of $A_{\infty}$ with respect to the state $\varphi_{\infty}$ results in a triplet $(\pi, \mathcal{H}, \Omega)$, where $\mathcal{H}$ is the constructed
Hilbert-space, $\pi$ is a $*$-homomorphism of $A_{\infty}$ into the set of all bounded operators $\mathcal{B}(\mathcal{H})$ on $\mathcal{H}$,
and $\Omega$ is the cyclic vector for $\pi$. We generate the von Neumann algebra $\mathcal{M}=\pi(A_{\infty})''$ by taking the bicommutant of $\pi(A_{\infty})$
and thus we make sure that all weakly or strongly convergent series have their limit point in $\mathcal{M}$.

Just like in Ref.~\cite{Ohya} we are going to denote by $\gamma$ the
right shift on $A_{\infty}$ which is defined for $x_1 \otimes x_2 \otimes \dots \otimes x_n \in M_{d^n}$ as
\begin{equation}
 \gamma(x_1 \otimes x_2 \otimes \dots \otimes x_n)=I\otimes x_1 \otimes x_2 \otimes \dots \otimes x_n \in M_{d^{n+1}}. \nonumber
\end{equation}
As a result, $\rho_n$ can be
written as
\begin{equation}
\rho_n=\left(\rho^{1/2}\right)^{\otimes n}\left( \sum_{i=0}^{n-1} a_{n,i+1}\gamma^i(X)
\right) \left(\rho^{1/2}\right)^{\otimes n} \nonumber
\end{equation}
It is obvious that $\varphi_{\infty}$ is invariant under $\gamma$. The extension $\varphi$ of the state $\varphi_{\infty}$
to $\mathcal{M}$ is
\begin{equation}
 \varphi(a)=\langle \Omega, a \Omega \rangle, \,\, a \in \mathcal{M}, \nonumber
\end{equation}
and the right shift $\gamma$ can be also extended such that $\varphi \circ \gamma = \varphi$. Now, the only
missing link in this approach is the connection between the relative entropy $S(\rho_n||\rho^{\otimes n})$ and the state $\varphi$.
Let us recall the Belavkin-Staszewski entropy and its relation to the relative entropy (see Ref.~\cite{Petz} and Proposition 7.11 in Ref.~\cite{Ohya})
\[
  S\left(\rho_n||\rho^{\otimes n}\right) \leqslant
  S_{BS}\left(\rho_n||\rho^{\otimes n}\right)
  =\mathrm{Tr} \Big\{\rho^{\otimes n} \eta\left((\rho^{-1/2})^{\otimes n} \rho_n (\rho^{-1/2})^{\otimes n} \right)  \Big \}
\]
where $\eta(t)=-t\log t$ and $\mathrm{supp}(\sigma) \subseteq \mathrm{supp}(\rho)$. One can simply realize that the right hand side
of the above equation is nothing else than the state $\varphi_n$ and therefore with the help of the GNS construction
we get
\begin{equation}
 S\left(\rho_n||\rho^{\otimes n}\right) \le \inn{\Omega,\eta\left( \sum_{i=0}^{n-1} a_{n,i+1}\gamma^i(\mathcal{X}) \right)\Omega}, \label{todo}
\end{equation}
where $\Omega$ is the cyclic vector and $\mathcal{X} \in \mathcal{M}$ is the embedding of $X=\rho^{-1/2}\sigma\rho^{-1/2}$ into $\mathcal{M}$.

\section{Proof of Theorem \ref{thm1}}

We would like to prove first that $\sum_{i=0}^{n-1}
a_{n,i+1}\gamma^i(\mathcal{X})\to \id_{\mcal{H}}$ strongly. In order to do that we need the
following theorem
(Theorem \ref{thm4}) from
Ref.~\cite{Jardas} (see Ref.~\cite{Yosida} pg.~213 for the
original theorem). For convenience of the reader we repeat a slightly modified
version of the statement here.

\begin{thm}\label{thm4}
  Let $Y$ be a Banach space and let $T:Y\to Y$ be a continuous linear operator
  such that the family of operators $\{T^n,n\in \mathbb{N}\}$ is power bounded. Further, let $A=(a_{n,i})(n,i\in \mathbb{N}^+)$ be a strongly regular matrix and
\[
 T_n=\sum_{i=1}^\infty a_{n,i}T^i,\quad (n\in \mathbb{N}^+).
\]
Then the strong limit $T_0=\lim_{n\to \infty} T_n$ exists and it is a projection on $Y$ onto the subspace $\mcal{F}=\{y\in Y:Ty=y\}$ of all
  fixed points of $T$. Moreover, projection $T_0$ does not depend on the choice
  of the strongly regular matrix $A$.
\end{thm}

The GNS construction provides us with a Hilbert space $\mcal{H}$ which is a
reflexive Banach space
and for the rest of the paper we are going to adopt the proof of Proposition
1.17 in Ref.~\cite{Ohya}. Let $V:\mcal{H}\to \mcal{H}$ be defined by
\begin{equation}
Va\Omega= \gamma(a)\Omega, \quad \forall a\in\mcal{M}. \nonumber
\end{equation}
The map $V:\mcal{H}\to \mcal{H}$ is an isometry on $\mcal{A}_\infty \Omega$:
\begin{multline*}
\inn{Va\Omega,Vb\Omega}= \inn{\gamma(a)\Omega,\gamma(b)\Omega} = \inn{(I\otimes
  a)\Omega, (I\otimes b)\Omega}\\ = \inn{\Omega, (I\otimes a^*)(I\otimes b)
  \Omega} = \inn{\Omega, (I\otimes a^*b)\Omega}= \varphi_\infty(I\otimes
  a^*b)\\ = \varphi_{\infty}(I)\varphi_\infty(a^*b) = \varphi_\infty(a^*b)=
  \inn{\Omega, a^*b\Omega} = \inn{a\Omega, b\Omega}.
\end{multline*}
Here, we used the fact that $\varphi_{\infty}\restriction_{M_{d^n}}=\varphi_n$.
Since $V$ is an isometry and $\overline{\pi(\mcal{A}_\infty)\Omega}=\mathcal{H}$,
$\{V^n,n\in\N\}$ is a power bounded family on
$\mcal{H}$, which means that we can apply Theorem \ref{thm4}
to our situation, where $A=(a_{i,j})$ is a strongly regular matrix. The isometry $V$ plays the role of $T$, hence
we need to find the fixed point subspace of $V$. Since the matrix $A$ does not
play any role in finding this subspace, the proof of Proposition 1.17 in
Ref.~\cite{Ohya} works verbatim and therefore the dimension of $\mcal{F}$ is one. Since
\begin{equation}
V\id_{\mcal{A}_\infty}\Omega=\gamma\left(\id_{\mcal{A}_\infty}\right)\Omega=
\gamma(I^{\otimes\infty})\Omega= I^{\otimes\infty}\Omega= \id_{\mcal{A}_\infty}\Omega, \nonumber
\end{equation}
the $\Omega$ is in $\mcal{F}$ and hence $\mcal{F}=\C\Omega$.

If $b'\in\mcal{M}'$, where $\mathcal{M}'$ is the commutant of $\mathcal{M}$, and $S_n(a)=\sum_{i=0}^{n-1} a_{n,i+1}\gamma^i(a)$,
 $a \in \mathcal{M}$, then
\[
S_n(a)b'\Omega=b'S_n(a)\Omega=b'\sum_{i=0}^{n-1} a_{n,i+1}\gamma^i(a)\Omega
=b'\sum_{i=0}^{n-1} a_{n,i+1}V^ia\Omega \to b'Ea\Omega,
\]
by Theorem \ref{thm4}, where $E$ is the projection of $\mcal{H}$
 onto $\mcal{F}$: $E(\xi)=\inn{\Omega,\xi}\Omega$ for all $\xi\in \mcal{H}$.
Thus
\[
b'Ea\Omega = b'\inn{\Omega,a\Omega}\Omega= \varphi(a)b'\Omega.
\]
We know from the proof of Proposition 1.17 in Ref.~\cite{Ohya} that
$\Omega$ is cyclic for $\mcal{M}'$, as well, hence $S_n(a)\to
\varphi(a)\id_{\mcal{H}}$ strongly. If we choose $a=\mathcal{X}$ then
\[
\varphi(\mathcal{X})=\mathrm{Tr} \{\rho \rho^{-1/2} \sigma \rho^{-1/2}\}=1.
\]
Hence $S_n(\mathcal{X})\to \id_{\mcal{H}}$ strongly.

We can finish the proof now.
Operator $\mathcal{X}$ is positive and the function $\eta(t)=-t\log t$ is
continuous on $[0,\infty)$. Since the continuous functional calculus preserves
the strong convergence, we obtain that $ \lim_{n \to \infty} \eta
\left(S_n(\mathcal{X}) \right)=0$ strongly. Therefore, with the aid of Eq.~\eqref{todo}
\[
\lim_{n \to \infty} \inn{\Omega,\eta\left( \sum_{i=0}^{n-1} a_{n,i+1}\gamma^i(\mathcal{X}) \right)\Omega}=0 \quad
 \Rightarrow \quad \lim_{n \to \infty} S\left(\rho_n||\rho^{\otimes n}\right)=0
\]
and the proof is complete.

Similarly, one can also obtain for
\[
 \rho_{n,k} = a_{n,1} \sigma^{\otimes k} \otimes \rho^{\otimes (n-k)}+ a_{n,2} \rho \otimes \sigma^{\otimes k} \otimes \rho^{\otimes (n-k-1)}

 +\dots+ a_{n,n-k+1} \rho^{\otimes (n-k)} \otimes \sigma^{\otimes k}
\]
with $\mathcal{X}=\rho^{-1/2}\sigma^{\otimes k}\rho^{-1/2}$, $\sum^{n-k+1}_{i=1} a_{n,i}=1$ and fixed $k$ the limit relation
\begin{equation}
 \lim_{n \to \infty} S(\rho_{n,k}||\rho^{\otimes n})=0. \nonumber
\end{equation}
This result extends a similar statement made in Ref.~\cite{Petz} and
further extends the set of states identified in Theorem \ref{thm1}.

\section*{Acknowledgement}

This work was supported by the Deutscher Akademischer Austauschdienst (M\"OB-DAAD project no. 65049).

\nocite{*}

redpajama_set_name: RedPajamaArXiv


text:
Suffolk slip to defeat in Mansfield
Published: 4:00 PM March 1, 2007
SUFFOLK returned from Mansfield in Nottinghamshire after suffering a 21-15 defeat in February's British Inter-Counties Championship darts fixture.The weekend started brightly, with the ladies B team recording a 4-2 victory, meaning they've still only lost one game this season.
SUFFOLK returned from Mansfield in Nottinghamshire after suffering a 21-15 defeat in February's British Inter-Counties Championship darts fixture.
The weekend started brightly, with the ladies B team recording a 4-2 victory, meaning they've still only lost one game this season. The winners were Issy Hurren (2-3, 14.99 av.), Nicky Doidge (1-3, 15.14 av.), Sylvia Rogers (1-3, 15.78 av.) and lady of the match Sue Thompson (2-3, 16.01 av.).
Things didn't go according to plan for the men's B team, who were defeated 8-4. The winners were James Philpot (1-3, 21.19 av.), Cavan Thake (1-3, 19.64 av.), Andy Trent (1-3, 21.88 av.) and, in his first appearance of the season and man of the match was Micky Rush (2-3, 25.19 av.).
Suffolk have grown used to an overnight score of 10-8 this season, so everyone was comfortable going into Sunday's A team matches. Once again, the day started brightly with the ladies A team recording a 4-2 victory, meaning they too have only lost once this season.
The winners were Louise Groves (0-3, 17.28 av.), Lady of the Match Harriet Wolton (1-3, 19.68 av.), Nicky Bradley (0-3, 17.68 av.) and captain Sue Talbot (2-3, 18.96 av.). Nicky Bradley also took the highest shot out purse on the day with 114.
The points were all square moving onto the men's A team match, but unfortunately, they crashed out 9-3 to better opposition on the day. The winners were Daniel Wilimott (2-3, 26.67 av.) who secured his game with a 110 shot out in the fifth leg, man of the match Steve Warren (0-3, 27.83 av.) and the team's most consistent player this season, Nicky Bloom (0-3, 21.17 av.).
With eight points from the ladies and just seven from the men, Nottinghamshire got the overall result 21-15 fair and square. The Suffolk team must all pull together and support each other during the remaining matches and lift themselves above the disappointment of this game.
IN the British Inter-County Youth, Northern Section on Sunday Febr-uary 18 saw the re-match between two of the most closely matched sides in the division - Suffolk and Lincolnshire. As some of the Lincolnshire party had Super League commitments in the evening, this match got underway promptly.
It was at 12 noon that Ollie Leggett looked to continue his unbeaten run as the U18 men's matches began.
Ollie didn't let the side down as he cruised to another 2-0 victory.
Wins followed for two more of Suffolk's brightest young players, namely Will Stan-more and Andrew Ward and Suffolk took a commanding 3-0 lead.
Lincolnshire took one back through Dale Thorpe, before Sean Lippett earned himself yet another MOM award with a fine 22.27 average in an apparently effortless performance!
With the section win safe, Suffolk eased up allowing Lincolnshire back into the match and they took the last two points.
Suffolk's U18's have been on fine form this season - keep it up team!
The match moved into the U21 ladies section and Suffolk stalwart Katie Welham, produced some fine scoring as she recorded a 2-0 win, and her 15.42 average was enough for the LOM award.
Harriet Wolton was next on, and due to some indifferent form on double one, she suffered a very rare defeat.
With all to play for, Hannah Rampley displayed some cool finishing against a promising looking newcomer to the Lincolnshire side, Velna Oldroyd, and made sure of a 2-1 victory for the ladies.
And so the overall score stood at 6-4, meaning a 3-2 loss by the U21's would be enough to secure the match.
SUFFOLK'S senior team face Wilthsire at home on March 3 and 4. The players toe-the-oche in the B team matches at 12noon on Saturday and the A teams commence battle at 11am on Sunday. Admission is £3 (which covers both days) and all spectators are welcome.The youth team are home to Bedfordshire on Sunday, March 11. This match will begin at 12.30pm and admission is just £1. Again, all spectators are welcome and if any youngsters are interested in finding out more about playing the sport of darts, they can speak to James, Keith or any of the other Suffolk officials and find out more.
Both matches take place at the home of Suffolk darts, The Riverside Snooker & Social Club, Stowmarket (tel: 01449 675571).

redpajama_set_name: RedPajamaCommonCrawl


text:
Gefco reports 'satisfactory results'; pushes on in Russia
By Automotive Logistics2015-04-22T12:24:19+01:00
Gefco has released financial results for 2014 that show a turnover of €4.1 billion ($4.4 billion), a 1.5% rise in turnover on the previous year. According to the transport and logistics provider current operating profit stands at €105m, a 10.5% increase on 2013.
Gefco executives highlighted the strength of the results in spite of several weak economic factors. However, the revenue growth was well short of its target for €4.5 billion this year. Last year, Gefco announced a target of €8 billion by 2020.
"Gefco achieved satisfactory results in the past year, considering the current challenging climate," said Luc Nadal, chairman of the Gefco management board (pictured).
The company did release yet its net profit figures for 2014, citing a special item that needed to be accounted for. In 2013, net profit was €58m.
Gefco was owned by PSA until 2012, when the French carmaker sold 75% of its stake in the logistics provider to RZD Russian Railways. PSA remains the company's largest customer, as Gefco maintains a global, dedicated contract for most of PSA's logistics operations.
In 2012 Gefco also signed a contract to be the dedicated fourth party logistics provider (4PL) to General Motors in Europe, including Russia.
Despite its pedigree and significant services across many aspects of automotive logistics, the company is clearly making efforts to diversify services beyond the sector as well as its two largest customers. Nadal pointed to 10% growth among its industrial logistics business outside of PSA and General Motors.
According to Gefco, it had expanded its portfolio of market clients by 10.6%. Non-automotive clients made up 29% of revenue in 2013. It has signed major contracts with Panstar and World Duty Free Group in Spain, for example, and with Turkish dental equipment maker Sirona Dental. It has also recently signed a contract with global bearing and industrial parts supplier Timken in Romania.
The company also made gains in other sectors, including electronics, consumer goods, aeronautics and industrial goods.
That said, Gefco reported that it had maintained its number one status in Europe for finished vehicle logistics, though no figures on the revenue generated from this sector were made available. However, Gefco would have benefited from a return to growth in the European Union, where PSA's sales increased 3.7% to 1.39m; while GM's overall sales in Europe fell 4%, mainly following the pullout of its Chevrolet brands in the region, Opel/Vauxhall sales were up 7% to 885,000.
Sales in the first quarter of 2015 have shown a stronger recovery in Europe, up 8.5% compared to last year. This past week, PSA announced that it would temporarily increase output in France by 10%, a move that will directly benefit Gefco.
Things have been more difficult in Russia recently, however, following a downturn last year that has now turned into a severe crisis. GM, which has been Gefco's biggest customer in the country, has already announced that it would pull most manufacturing out of the country and Peugeot Citroën's joint venture with Mitsubishi – PCMA Rus – has idled production between April and July.
Those decisions followed PSA's move to cancel rail shipments of parts and CKDs from France and wider Europe to the Kaluga facility, which Gefco handled.
Despite these setbacks, Gefco has continued to look at other opportunities in Russia together with its major shareholder. The company said it had sped developments in Russia and the Commonwealth of Independent States (CIS) by creating a dedicated geographic zone – the 1520 zone, which includes Russia, Ukraine, Kazakhstan and the Baltic states. To look after this zone the company has formed a cluster of 50 logistics experts based in Moscow. The company said that its diversification strategy had been particularly successful in the 1520 zone, with 350 new contracts signed in 2014.
It has also established a door-to-door rail transport service between Asia and Europe connecting the Chinese rail network with Russia as well as countries including Belarus, Germany, Hungary, Kazakhstan and Uzbekistan.
As reported in February Gefco has seen growing automotive interest for a Europe-Asia rail service. Since Russian Railways (RZD) took majority control of Gefco in 2012, the companies have been targeting trade between Europe and China, including across the Trans-Siberian Railway, with Gefco expected to use its global network of customers to drive traffic across RZD's infrastructure.
Elsewhere in its business, Gefco has also seen difficulties in its French groupage business, a segment that has shown consistent losses. Nadal said during a press conference that Gefco would cut around 500 jobs in the segment, without closing any depots.
Reporting by Christopher Ludwig and Marcus Williams
Integrated LSPs
Shipping - vehicles


For Bozz Media Web Hosting clients. If you do not use Google Apps or MS Exchange server, you may end up utilizing email forwarding addresses via your webserver, such as yourname@yourdomain.com, to both send and receive email with your gmail account (ex: bozzmedia.client@gmail.com). This is a walk through for setting that up.
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Select 'Add another email address you own'. Additional notes in the screenshot to consider after you have successfully setup your new address.
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When you write a new email or reply, you can select which email you are using in the From field.
The absolute easiest way to setup your custom email on your mobile phone is to use the Gmail App.
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text:
\section{Introduction}

Chaotic systems exhibit a wealth of quantum phenomena.
A canonical example is the quantum kicked rotor~\cite{Casati1} (see
Refs.~\cite{Fishman10, Izrailev90, Chirikov79} for reviews)---a free
 rotating particle under the influence
of sequential time-periodic driving, the Hamiltonian of which reads
\begin{eqnarray}
\hat{H}=\frac{1}{2}(\tilde h\hat n)^2 + K\cos\hat{\theta} \sum_{m}\delta(t-m).
\label{eq:50}
\end{eqnarray}
Here the time $t$ is rescaled by the kicking period $\tau$, the Planck's
quantum $\tilde h=\hbar\tau/I$ with $I$ the particle's moment of inertia,
and the angular momentum $\hat n$ canonically conjugates to the angular
position $\theta$. The dimensionless amplitude of the kicking potential,
$K$, namely the so-called classical stochastic parameter, governs the
degree of the system's nonlinearity.
Despite this seemingly simple construction, tuning $\tilde h$ gives rise
to rich dynamical behaviors. For example, for (generic) irrational
values of $\tilde h/(4\pi)$ the rotor's kinetic energy saturates at long
times~\cite{Casati}, in sharp contrast to the linear energy growth in the
classical limit of vanishing $\tilde h$~\cite{Chirikov}. The former, the
so-called dynamical localization, simulates an Anderson insulator
\cite{Fishman} while the latter simulates a (normal) metal. Even more  striking,
$\tilde h$-driven phenomena occur to variants of the kicked  rotor
and they simulate a broad spectrum of condensed-matter systems (e.g.,
Refs.~\cite{KH1, KH2, KH3, KH4, KH5, Gong12, Altland11, Wang11, Gong15}).
Most recently, it has been found that for a spinful quasiperiodically
kicked rotor, the Planck's quantum can drive a sequential rotor Anderson
insulator-metal transition shown to be the mathematical equivalence of
the integer quantum Hall effect~\cite{Chen}.

A peculiar phenomenon common to various kicked rotor systems is the so-called
quantum resonance for rational values of $\tilde h/(4\pi)=p/q$, with $p$ and $q$ being
coprime natural numbers~\cite{QR}. For these values of $\tilde h$ the rotor's
kinetic energy grows quadratically at large times, which is much faster than
the metallic (linear) growth. Such a phase simulates a perfect crystal with
diverging electric conductivity corresponding to ballistic electron motion.
Therefore, it is dubbed ``supermetal''. Physically, for rational $\tilde h/(4\pi)$
the system exhibits translation symmetry in the angular momentum space, and
resonant transmission through the ensuing Bloch bands gives rise to the
supermetallic (quadratic) growth. The extreme sensitivity of the system's behavior
such as dynamical localization and quantum resonance to the number-theoretic
property of $\tilde h/(4\pi)$ is a prominent feature that distinguishes the
kicked rotor from genuine disordered systems (see Refs.~\cite{Tian, Altland11,
Casati, Dana, Wimberger, Izrailev90, Guarneri09, Fishman03a} for reviews of
the early and recent status of this subject). Over decades, this subject has
been central to theoretical and experimental studies of quantum chaos. Recently,
the supermetallic phase of the kicked rotor has been experimentally observed
in chemical systems and finds potential applications such as achieving
spin-selective rotational excitation of molecules~\cite{Milner12}.

So far, most studies have focused on the asymptotic supermetallic growth and
are mute to effects of chaoticity. In Ref.~\cite{Tian}, it is first found by
using the field theory that for the standard kicked rotor the chaoticity gives
rise to a universal crossover from metallic to supermetallic energy growth
which is insensitive to the system's details such as the values of $p$ and $q$ and
the strength of the kicking potential. Moreover, it has been found that this
universal dynamical behavior has a deep connection to the optical conductivity
of perfect crystals~\cite{Taniguchi93, Tian09} and the quantum walk in the
periodic multi-baker map~\cite{Dorfman1, Dorfman2, Dorfman}. A fundamental
question naturally arises: How robust is this dynamical behavior? Notably,
it is unclear whether such dynamics universality exists in general periodic
chaotic systems and is impervious to the symmetry. The present work aims at
a systematic study of these issues.

We study the supermetallic phase (rational $\tilde h/(4\pi)$) of a large
class of generalized kicked rotor systems. These systems have qualitatively
different dynamical behaviors (e.g., dynamical localization or delocalization)
when $\tilde h/(4\pi)$ is irrational. We analytically show and numerically
confirm that chaoticity leads to even richer universal behavior in the
crossover from metallic to supermetallic energy growth. We find that the
universal dynamics crossover behavior is very sensitive to the symmetry
of the Hamiltonian describing the free rotation of the particle, but not
to system's details. We show that the universal metal-supermetal dynamics
crossover can be attributed to the universality of fluctuations of the Bloch
wave functions and bands of the reduced quantum system in a unit cell. The
latter motivates us to conjecture that the universality class of supermetal
dynamics is identical to that of eigenfunction and spectral fluctuations
described by the random matrix theory (RMT)~\cite{RMT}.

The rest of the paper is organized as follows. In the next section we
introduce the models and discuss their symmetries. We present the analytical
results of the energy growth in Sec.~\ref{sec:fieldresults} and put them to
numerical test in Sec.~\ref{sec:simuresults}. In Sec.~\ref{sec:discussion}
we discuss the RMT foundation of the universality of supermetal dynamics.
We conclude in Sec.~\ref{sec:conclusion}. Some technical details are
presented in Appendix~\ref{sec:derivation}.

\section{\label{sec:Floquet} Model and symmetry}

In this work we will explore the dynamical behavior of a large class of
generalized quantum kicked rotors for rational values of $\tilde h/(4\pi)$.
In this section we first describe the models and discuss their symmetries.

\subsection{Generalized quantum kicked rotor}
\label{sec:KR}

The system to be considered below has a general Hamiltonian as follows,
\begin{eqnarray}
\hat{H}=H_0(\hat n) + K\cos\hat{\theta} \sum_{m}\delta(t-m).
\label{eq:13}
\end{eqnarray}
Here we assume the free rotation
Hamiltonian $H_0 (\hat n)$ to be an analytic function of $\hat n$ which can
be expressed as
\begin{equation}\label{eq:10}
H_0 (\hat n)=\sum_{k=0}^\infty c_k \hat n^k,
\end{equation}
where the numerical coefficients $c_k$ generally depend on $\tilde h$.
The angular momentum operator $\hat n$ canonically conjugates to the angular
operator $\hat \theta$ and has the eigenvalue spectrum $\{\tilde n\}, \tilde
n\in \mathbb{Z}$. When $H_0 (\hat n)=\frac{1}{2}\tilde h^2\hat n^2$ the system
is reduced to the standard quantum kicked rotor (\ref{eq:50}). The quantum
evolution can be expressed as a stroboscopic dynamics such that the wave
vector at integer time $t$, $| \psi(t) \rangle$, is given by
\begin{eqnarray}
| \psi(t) \rangle = \hat{U}^t | \psi(0)\rangle, \quad t\in \mathbb{Z},
\label{eq:6}
\end{eqnarray}
with the Floquet operator
\begin{eqnarray}
\hat{U}=e^{-\frac{i}{2\tilde h}H_0}e^{-\frac{iK}{\tilde h}\cos\hat\theta}
e^{-\frac{i}{2\tilde h}H_0}.
\label{eq:4}
\end{eqnarray}
As mentioned above, we shall focus on $\tilde h/(4\pi)=p/q$ throughout
this work.

\subsection{Symmetry}
\label{sec:symmetries}

First of all, we shall consider such $c_k$ that $H_0/\tilde h$ is shifted
by multiple $2\pi$ upon the translation operation
\begin{equation}\label{eq:11}
\hat T_q: \quad \hat n\rightarrow \hat n+q.
\end{equation}
As a result,
\begin{eqnarray}{\label{eq:12}}
[\hat{U},\hat{T}_q]=0,
\end{eqnarray}
implying that the system exhibits the translation symmetry in the
$\hat n$ space. This brings the rotor to the supermetallic phase.

Next, the system exhibits an effective time-reversal symmetry. To be
specific, we note that the Hamiltonian (\ref{eq:13}) is invariant under
the `time-reversal' operation
\begin{equation}\label{eq:14}
\hat T_c: \quad \hat n\rightarrow \hat n,\, \hat \theta\rightarrow -\hat
\theta, \, t\rightarrow -t.
\end{equation}
This has an important consequence, i.e.,
\begin{equation}\label{eq:15}
 \hat U^{\cal{T}}=\hat U,
\end{equation}
where the superscript `$\cal{T}$' stands for the transpose. This $\hat T_c$
symmetry brings the system to the orthogonal class in the RMT.

Finally, if $c_k=0$ for all odd $k$, then $H_0(\hat n)$ and thereby the
system bears an additional symmetry, i.e., $\hat U$ being invariant under
the operation
\begin{eqnarray}
\hat{T}_i: \quad \hat n \rightarrow -\hat n.
\label{eq:2}
\end{eqnarray}
Otherwise this symmetry is broken. This symmetry is dubbed the ``inversion
symmetry'' following solid-state physics \cite{Mermin}. Its effects on the
quantum resonance have not yet been explored~\cite{note1} and this is the
main subject of this work.

\section{\label{sec:fieldresults}Theory of supermetal dynamics}

In this section we present an analytic theory for dynamics of a generalized
quantum kicked rotor at resonance, i.e., the supermetallic phase. Particular
attentions are paid to effects of the inversion symmetry. Armed with this
theory, we explicitly calculate the rotor's `kinetic energy'~\cite{note2}
\begin{eqnarray}
E(t)\equiv\frac{1}{2} \langle \psi(t)|\hat{n}^2|\psi(t) \rangle.
\label{Etdf}
\end{eqnarray}
Throughout, to simplify technical discussions without loss of generality,
we assume that the initial state is an unperturbed eigenstate with zero
angular momentum, i.e., $|\psi(0)\rangle =|0 \rangle$.

\subsection{Reduced system}
\label{sec:reduction}

First of all, the quantum dynamics (\ref{eq:6}) in the angular momentum
space is determined by the quantum amplitude, $\langle \tilde n|\hat U^t|
\tilde n'\rangle$. It is easy to show that the time Fourier transformations
of this amplitude and its complex conjugate, denoted as $\langle \tilde
n|(1-e^{i\omega_+}\hat U)^{-1}|\tilde n'\rangle$ and $\langle \tilde n'|
(1-e^{-i\omega_-}\hat U^\dagger)^{-1}|\tilde n\rangle$, respectively,
give
\begin{eqnarray}
    E(t)&=&\frac{1}{2}\sum_{\tilde n} \tilde n^2 \int \frac{d\omega}{2\pi}
    e^{-i\omega t}\label{eq:21}\\
    &\times& \langle\! \langle \tilde n|(1-e^{i\omega_+}\hat U)^{-1}|0\rangle
    \langle 0|(1-e^{-i\omega_-}\hat U^\dagger)^{-1}|\tilde n\rangle\!
    \rangle_{\omega_0},
    \nonumber
\end{eqnarray}
where $\omega_\pm=\omega_0 \pm \frac{\omega}{2}$ and $\langle \cdot
\rangle_{\omega_0}\equiv \frac{1}{2\pi}\int_0^{2\pi}d\omega_0$.

To proceed we consider the eigenvalues and eigenvectors of $\hat U$.
The translation symmetry (\ref{eq:12}) entails a good quantum number, i.e.,
the Bloch angle $\theta \in [0,2\pi/q]$. According to Bloch's theorem,
\begin{eqnarray}
\hat{U}|\psi_{\alpha,\theta} \rangle = e^{i \epsilon_\alpha(\theta)}|
\psi_{\alpha,\theta} \rangle,
\label{eq:3}
\end{eqnarray}
where both the quasienergy spectrum, $\{\epsilon_\alpha(\theta)\}$, and
corresponding eigenvectors, $\{|\psi_{\alpha,\theta} \rangle\}$, depend
on $\theta$. The Bloch wave function
\begin{equation}\label{eq:23}
\psi_{\alpha,\theta}(\tilde n)\equiv \langle \tilde n|\psi_{\alpha,\theta}
\rangle=e^{i\tilde n\theta}\varphi_{\alpha,\theta}(\tilde n),
\end{equation}
with $\varphi_{\alpha,\theta}(\tilde n)$ being a $q$-periodic wave function,
\begin{equation}\label{eq:20}
\varphi_{\alpha,\theta}(\tilde n)=\varphi_{\alpha,\theta}(\tilde n+q)=
\varphi_{\alpha,\theta}(n).
\end{equation}
In deriving the last equality of Eq.~(\ref{eq:20}) we use the fact that
$\tilde n$ can be uniquely written as $n+Nq$, where $0\leq n \leq q-1$ and
$N\in \mathbb{Z}$. Upon applying Bloch's theorem to Eq.~(\ref{eq:21})
we express the energy profile as
\begin{eqnarray}
E(t)=-\frac{1}{2}\int\frac{d\omega}{2\pi}e^{-i\omega t}{\sum_{n=0}^{q-1}}
\int_b d\theta_+ \partial_{\theta_-}^2|_{\theta_-=\theta_+}K_\omega(n),
\label{eq:16}
\end{eqnarray}
where the notation: $\int_b d\theta \equiv \frac{q}{2\pi}
\int_0^{2\pi/q} d\theta$ has been introduced and
\begin{eqnarray}\label{eq:22}
    K_\omega(n)&=&\langle\! \langle n|\frac{1}{1-e^{i\omega_+}
    \hat U_{\theta_+}}|0\rangle \nonumber\\
    &\times& \langle 0|\frac{1}{1-e^{-i\omega_-}\hat U^\dagger_{\theta_-}}
    |n\rangle\!\rangle_{\omega_0}
\end{eqnarray}
is the density correlation function for autonomous stroboscopic dynamics
restricted on a circle of circumference $q$. This reduced system is governed
by a $\theta$-dependent Floquet operator,
\begin{eqnarray}
\hat{U}_\theta = e^{-\frac{i}{2\tilde h}H_0}e^{-\frac{iK}{\tilde h}
\cos(\hat\theta+\theta)}e^{-\frac{i}{2\tilde h}H_0}.
\label{eq:5}
\end{eqnarray}
Physically, $K_\omega(n)$ probes interference between retarded and advanced
amplitudes corresponding to the reduced dynamics, and the Bloch angle $\theta$
introduces an ``Aharonov-Bohm flux'', $q\theta$, piercing through the circle
(see Fig.~\ref{fig:1}). Note that the unitarity of $\hat{U}_\theta$ now implies
$\sum_{n''=0}^{q-1}(\hat{U}_\theta)_{nn''}(\hat{U}_\theta^\dagger)_{n''n'}=
\delta_{nn'}$ for arbitrary $n,n'\in \{0,1,\cdots, q-1\}$.

It is important to note that for the reduced system governed by
$\hat{U}_\theta$ the effective time-reversal symmetry $\hat T_c$ is broken
by the Bloch angle if $\theta\neq 0,\,\pi$. A question naturally arises: Does
this broken $\hat T_c$ symmetry bring the system to the unitary class in the
RMT? As we will show in the following, the answer crucially depends on whether
$H_0(\hat n)$ exhibits the inversion symmetry $\hat T_i$. If the latter
symmetry is broken, then the system belongs to the unitary class. If not,
then $\hat{U}_\theta$ is invariant under the combined operation $\hat T_i
\hat T_c$ and this brings the reduced system back to the orthogonal class.

\begin{figure}
\includegraphics[width=8.5cm]{fig1.eps}
\caption{(Color online) The translation symmetry entails a reduction of
quantum dynamics from unbounded angular momentum space to a circle of
circumference $q$ pierced by an Aharonov-Bohm like flux $q \theta$ with
$\theta$ being the Bloch angle.}
\label{fig:1}
\end{figure}

\subsection{Universal metal-supermetal dynamics crossover}
\label{sec:energy_growth}

\subsubsection{Field theory}
\label{sec:field_theory}

The remaining task is to calculate the two-particle Green's function
${K_\omega(n)}$ evaluated at fixed parameters $\theta_\pm$. Following
Ref.~\cite{Tian} we may express it in terms of a functional integral,
\begin{eqnarray}
K_\omega(n)&=&-\frac{1}{2^4} \int dQ\, e^{-S[Q]}\nonumber\\
&\times& {\rm str}\left(k
(Q(n))_{+2,-2}k(Q(0))_{-2,+2}\right).
\label{eq:17}
\end{eqnarray}
Here $Q\equiv \{Q_{\lambda \alpha \beta,\lambda' \alpha' \beta'}\}$ is
a supermatrix defined on three sectors: the index $\lambda=+,-$ refers
to the advanced and retarded (`AR') sector, $\alpha={f,\, b}$ to the
fermionic-bosonic (`FB') sector, and $\beta=1,2$ to the `T' sector
accommodating the symmetry under the combined operation $\hat T_i\hat T_c$;
i.e., $\beta=1$ ($2$) refers to (in)variance under the $\hat T_i\hat T_c$
operation. $Q$ satisfies the periodic boundary condition $Q(n)=Q(n+q)$.
The constant matrix $k=\sigma_{\rm FB}^3\otimes \sigma_{\rm AR}^0 \otimes
\sigma_{\rm T}^0$, where $\sigma_{\rm X}^0$ is the unit ($2\times 2$)
matrix in the X sector. The action is given by
\begin{eqnarray}
\label{eq:18}
  S = \frac{1}{2^4}\int_0^q dn {\rm str}\big(D_q(i\partial_n
  Q-[\hat \vartheta,Q])^2-2i\omega Q\sigma_{\rm AR}^3\big)
\end{eqnarray}
with
\begin{equation}\label{eq:19}
    \hat \vartheta=\left(
         \begin{array}{cc}
         \theta_+ & 0 \\
         0 & \theta_- \\
         \end{array}
   \right)_{\rm AR}\otimes \sigma_{\rm T}^{-1+\beta^2}\otimes
   \sigma_{\rm FB}^0.
\end{equation}
For $\beta=2$ the matrix $\sigma_{\rm T}^{-1+\beta^2}=\sigma_{\rm T}^3$
represents the sign change in the Aharonov-Bohm flux under the $\hat T_c$
operation. The corresponding effective field theory has been obtained
in Ref.~\cite{Tian}, but for $\beta=1$, an additional symmetry,
\begin{equation}\label{eq:33}
    Q(n)=Q(q-n),
\end{equation}
arises.
Note that the above action is universal in the sense that
it depends on system parameters only through the diffusion constant
$D_q$. The effective field theory is valid, provided the parametric
conditions, (i) $1\ll K/\tilde h \ll q \ll (K/\tilde h)^2$, (ii) $\omega
\ll 1$, and (iii) $K\gg 1$ \cite{note5}, are met. The physical meanings
of these conditions are as follows. For (i), $K/\tilde h$ plays the role
of the ``transport mean free path'' in normal metals; the inequality
$K/\tilde h \ll q$ guarantees the validity of the hydrodynamic expansion,
while $q \ll (K/\tilde h)^2$ implies that localization physics does not
play any roles. For (ii), means that we are concerned in time scales much larger
than the kicking period. In (iii) and $K/\tilde h\gg 1$, it is implied that the angular
(or more precisely, $\sin\hat\theta$) correlation rapidly decreases, i.e.,
strong chaoticity. The latter results in the fact that the distribution of the
quasienergy spectrum $\{\epsilon_\alpha(\theta)\}$ of the Floquet operator
$\hat{U}_\theta$ follows the Wigner-Dyson statistics for the circular
orthogonal ensemble (COE) for $\beta=1$ and the circular unitary ensemble
(CUE) for $\beta=2$.

In principle, the diffusion constant $D_q$ depends on both the parameters
$K$ and $\tilde h$ and the details of $H_0$, i.e., $c_k$. The $H_0$ dependence
arises from the short-time correlation contributions to the diffusion
constant~\cite{Tian}. For $K/\tilde h \gg 1$ these correlation contributions
are negligible and consequently only the parameter ($K,\tilde h$) dependence
arises, i.e., $D_q = D_q(K, \tilde h)\sim(K/\tilde h)^2$.

\begin{figure}
\includegraphics[width=8.7cm] {fig2.eps}
\caption{(Color online) The analytical results obtained by the field
theory for a generalized kicked rotor with (thick blue dashed line) and
without (thick red dotted line) the inversion symmetry. It shows that the
energy profile exhibits a universal crossover from metallic ($\sim \tilde
t$) to supermetallic ($\sim \tilde t^2$) growth. The results are in excellent
agreement with those obtained from the RMT corresponding to the orthogonal
(thin green solid line) and the unitary (thin black dash-dotted line)
symmetry, respectively.}
\label{fig:2}
\end{figure}

\subsubsection{Metallic-supermetallic growth crossover}
\label{sec:explicit_results}

We proceed to use Eqs.~(\ref{eq:16}), (\ref{eq:17}), and (\ref{eq:18}) to
explicitly calculate $E(t)$. First of all, for $\omega\lesssim \Delta\equiv
2\pi/q$ which is the mean level spacing of the unit cell, the effective action
(\ref{eq:18}) shows that inhomogeneous $Q$-field fluctuations have a typical
action $E_c/\Delta \gg1$ where the Thouless energy $E_c = D_q/q^2$ is the
inverse of the classical diffusion time through the periodicity volume. This
inequality is ensured by $q\ll (K/\tilde h)^2\sim D_q$. So, inhomogeneous
fluctuations can be neglected and only the zero mode, $Q(n)\equiv Q$, is kept.
Equation (\ref{eq:18}) thereby is simplified to the zero mode action,
\begin{eqnarray}
S=\frac{\pi}{8\Delta}{\rm str}(D_q[\hat{\vartheta},Q]^2-2i\omega Q\sigma_{\rm AR}^3).
\label{eq:27}
\end{eqnarray}
Then, we substitute this action into Eq.~(\ref{eq:17}). The subsequent
calculations are similar to those of Ref.~\cite{Tian} and below we will give
the results. Some details are given in Appendix~\ref{sec:derivation}. We find
\begin{equation}
\label{eq:26}
    E(t)/(qD_q)=F(\tilde{t}),\quad \tilde{t}=t/q.
\end{equation}
It is important that this energy profile is universal: the details of the model
[e.g., the stochastic parameter $K$, the free Hamiltonian $H_0$, the denominator
of rational $\tilde h/(4\pi)$, etc.] only determine the scales of energy and
time. The explicit form of the universal function $F(\tilde t)$ depends on
the system's symmetry and is given as follows:

\begin{figure}
\vskip-0.038cm
\includegraphics[width=8.7cm]{fig3.eps}
\caption{(Color online) The simulation results of the distribution of the nearest
level spacing $P(s)$ for $\alpha_2=1,\alpha_3=0$ (green solid histograms)
and $\alpha_2=0, \alpha_3=1$ (black dash-dotted histograms) follow the Wigner-Dyson statistics for COE (blue dashed lines) and CUE
(red dotted lines), respectively. The parameter $\tilde h/(4\pi)$ is
$55/691$ (a), $24/301$ (b), $24/299$ (c), and $7/87$ (d). For all four panels
$K=300$.}
\label{fig:3}
\end{figure}

$\bullet$~For $H_0$ with inversion symmetry so that the quantum system reduced
to a unit cell is orthogonal, we find (see Appendix~\ref{sec:derivation} for
derivations)
\begin{eqnarray}\label{eq:7}
F\left(\tilde{t}\right)&=&\frac{1}{8}\int_1^\infty d\lambda_1\int_1^\infty d
\lambda_2\int_{-1}^1 d\lambda \delta(2\tilde{t}+\lambda-\lambda_1\lambda_2)
\nonumber\\
&\times& \frac{(1-\lambda^2)(1-\lambda^2-\lambda_1^2-\lambda_2^2+2\lambda_1^2
\lambda_2^2)^2}{(\lambda^2 + \lambda_1^2 + \lambda_2^2 - 2\lambda\lambda_1
\lambda_2-1)^2}.
\end{eqnarray}
For short times ($\tilde t\ll 1$) the Dirac function in Eq.~(\ref{eq:7})
implies that the integrals are dominated by $\lambda, \lambda_{1,2}\approx 1$.
Therefore, we may expand $\lambda,\lambda_{1,2}$ near unity and keep the leading
expansion of $F(\tilde t)$. As a result, $F(\tilde t\ll 1)\sim \tilde t$. For
long times ($\tilde t\gg 1$), the integrals are dominated by $\lambda={\cal O}
(1)$ and $\lambda_{1,2}\gg 1$. Taking this into account we find $F(\tilde t
\gg 1) \sim \tilde t^2$.

$\bullet$~For $H_0$ without the inversion symmetry so that the quantum system
reduced to a unit cell is unitary, the result is the same as what has been
found previously~\cite{Tian}, which is
\begin{eqnarray}\label{eq:8}
F\left(\tilde{t}\right)=\bigg\{
\begin{array}{rcl}
\tilde{t}+\frac{1}{3}\tilde{t}^3,&0<\tilde{t}<1,\\
\tilde{t}^2+\frac{1}{3},&\tilde{t}>1.
\end{array}
\end{eqnarray}

Eqs.~(\ref{eq:7}) and (\ref{eq:8}) show that the energy
profile exhibits a universal crossover from metallic to supermetallic
growth (Fig.~\ref{fig:2}). As we will show in Sec.~\ref{sec:discussion},
they can be obtained also from RMT.

\section{\label{sec:simuresults} Numerical tests}

In this section we put the analytic results,
namely, Eqs.~(\ref{eq:7}) and (\ref{eq:8}), to numerical tests. As we
will see below, although the analytical derivations of these two universal
crossovers require the conditions $1\ll K/\tilde h\ll q \ll (K/\tilde h)^2$
and $K\gg 1$, they hold in broader regimes. Surprisingly, they hold even in
systems (e.g., the kicked Harper system~\cite{KH1, KH2, KH3}) the quantum
dynamical behaviors of which at irrational $\tilde h/(4\pi)$ are fundamentally
different from the standard kicked rotor~\cite{KH4, KH5}.

To numerically study the universality of the dynamics crossover we
use different forms of the free Hamiltonian $H_0$. It has been known that
this Hamiltonian, $H_0$, mimics ``impurities'' in genuine disordered
systems~\cite{Fishman, Tian}.

\subsection{\label{sec:polynomial} Polynomial-type free Hamiltonian}

We first consider a polynomial-type free Hamiltonian with a specific form
given by
\begin{eqnarray}\label{eq:1}
H_0 = \frac{\tilde h^2}{2}\left(\alpha_2 \hat{n}^{2}+\alpha_3 \hat{n}^{3}\right),
\end{eqnarray}
with $\alpha_{2,3} \in \mathbb{N}$. The ${\hat T}_i$ symmetry is present
for vanishing $\alpha_3$ and otherwise broken.

\subsubsection{Spectral statistics}
\label{sec:spectral_statistics}

\begin{figure}[!]
\includegraphics[width=8.7cm]{fig4.eps}
\caption{(Color online) The simulation results of $P(s)$ (black dash-dotted
histograms) follow the Wigner-Dyson statistics for CUE (red dotted line), as long as $\alpha_3$ does not vanish. For
panel (a) $\alpha_2=1$ and $\alpha_3=99$ while for (b) $\alpha_2=99$ and
$\alpha_3=1$. In both cases $\tilde h/(4\pi)=8/101$ and $K=300$. The blue
dashed lines are for the Wigner-Dyson statistics for COE.}
\label{fig:4}
\end{figure}

For a given Bloch angle $\theta$ we numerically diagonalize the Floquet operator
$\hat{U}_\theta$ to obtain the quasienergy spectrum $\{\epsilon_\alpha(\theta)\}$.
We repeat the computation for $1000$ randomly selected values of $\theta$ so that
a large ensemble is realized. We calculate the distribution of the nearest
level spacing, $P(s)$, for various parameters of $\alpha_{2,3}, K$ and $\tilde h$.
We find that $P(s)$ follows the Wigner-Dyson statistics
for COE for vanishing $\alpha_3$ (Fig.~\ref{fig:3}, green solid histograms) and CUE for
vanishing $\alpha_2$ (Fig.~\ref{fig:3}, black dash-dotted histograms). More generally, as
long as $\alpha_3\neq 0$, $P(s)$ follows the Wigner-Dyson statistics for CUE (Fig.~\ref{fig:4}).
These results indicate that the system is strongly chaotic so that the field
theory is expected to be valid.

\begin{figure}
\vskip-0.038cm
\includegraphics[width=8.7cm]{fig5.eps}
\caption{(Color online) The simulation results of the rescaled energy
growth $F(\tilde t)$ for $\alpha_2=1,\alpha_3=0$ (green solid lines) and
$\alpha_2=0, \alpha_3=1$ (black dash-dotted lines) convincingly support
the analytic predictions for orthogonal (blue dashed lines) and unitary
(red dotted lines) systems, respectively. The parameters for each panel
are the same as in Fig.~\ref{fig:3}.}
\label{fig:5}
\end{figure}

\subsubsection{Universal energy growth}
\label{sec:diffusive-ballistic_crossover}

For a given free Hamiltonian $H_0$, we use the standard fast Fourier
transform technique to simulate the quantum evolution (\ref{eq:6}) to
compute $E(t)$ defined by Eq.~(\ref{Etdf}) and in turn $F(\tilde{t})$
by Eq.~(\ref{eq:26}).
Fig.~\ref{fig:5} presents the simulation results of $F(\tilde t)$ for
the free Hamiltonian $H_0$ given by Eq.~(\ref{eq:1}) with the same parameters
as in Fig.~\ref{fig:3} for the sake of comparison. We find that for vanishing
$\alpha_3$ where the ${\hat T}_i$ symmetry is present, $F(\tilde{t})$ is
in excellent agreement with the analytic result given by Eq.~(\ref{eq:7}),
while for vanishing $\alpha_2$ where the ${\hat T}_i$ symmetry is broken,
$F(\tilde{t})$ is in excellent agreement with the analytic result given by
Eq.~(\ref{eq:8}). We also find that the universal metallic-supermentallic
growth crossovers (\ref{eq:7}) and (\ref{eq:8}) are valid not only in the
regime of $K/\tilde h\ll q \ll (K/\tilde h)^2$ (Fig.~\ref{fig:5}(a)), but
also of $q \lesssim K/\tilde h$ (Fig.~\ref{fig:5}(b)-(d)). In
addition, as shown in Fig.~\ref{fig:6}, as long as $\alpha_3\neq 0$, for
which the quasienergy spectrum follows the Wigner-Dyson statistics for
CUE (see Fig.~\ref{fig:4}), the energy growth follows the
metallic-supermetallic growth crossover of unitary type described by
Eq.~(\ref{eq:8}), confirming the sensitivity of the crossover to
the system's symmetry.

\begin{figure}[!]
\includegraphics[width=8.7cm] {fig6.eps}
\caption{(Color online) The simulation results (black dash-dotted lines)
show that as long as $\alpha_3$ does not vanish, $F(\tilde t)$ follows
the metallic-supermetallic growth crossover of unitary type described
by  Eq.~(\ref{eq:8}) (red dotted lines). The parameters for each panel
are the same as in Fig.~\ref{fig:4}. For comparison, the crossover of
orthogonal type described by Eq.~(\ref{eq:7}) (blue dashed lines) is
also plotted.}
\label{fig:6}
\end{figure}

\subsection{\label{sec:KHM} Nonpolynomial-type free Hamiltonian}

When the polynomial is of higher degree (with all coefficients being
integers) but finite, we find numerically that the system's behavior at
resonance remains the same. A natural question thereby arises: What happens
if the degree is infinite so that $H_0$ is not a polynomial? To investigate
this problem we consider
\begin{eqnarray}\label{eq:9}
H_0=L\cos(\tilde h\hat{n}+\phi).
\end{eqnarray}
Here $L$ is a (nonzero) constant. When the phase parameter $\phi$ vanishes,
this system becomes the conventional kicked Harper model~\cite{KH1,KH2,KH3}
and exhibits the inversion symmetry. Whereas $\phi\neq 0$ the inversion
symmetry breaks.

As shown in Fig.~\ref{fig:7}, the spectral fluctuations of the Floquet
operator $\hat U_\theta$ obey the Wigner-Dyson statistics for COE (CUE) for $\phi=0$ ($\pi/2$). Correspondingly, $F(\tilde{t})$ is
in excellent agreement with the analytic result given by Eq.~(\ref{eq:7})
for $\phi=0$ (Fig.~\ref{fig:8}(a)) and by Eq.~(\ref{eq:8})
(Fig.~\ref{fig:8}(b)) for $\phi=\pi/2$ . Therefore, the analytic results
for universal metal-supermetal dynamics crossover are also confirmed for
the kicked Harper model.

\begin{figure}
\includegraphics[width=8.7cm] {fig7.eps}
\caption{(Color online) The simulation results of $P(s)$ for the
kicked Harper model. The green solid histogram for $\phi=0$ in (a)
and the black dash-dotted histogram for $\phi=\pi/2$ in (b) follow
the Wigner-Dyson statistics for COE (blue dashed line) and
CUE (red dotted line), respectively. For both panels
$\tilde h/(4\pi)=24/301$ and $L=K=300$.}
\label{fig:7}
\end{figure}

\section{\label{sec:discussion}Dynamics universality from RMT}

In this section we provide an alternative scheme for approximate analytic
calculations of the energy profile. From this scheme we will see that
the universality of the dynamics crossover is deeply rooted in the universal
fluctuations of the Bloch's bands and eigenfunctions which are well
described by the RMT. Interestingly, this scheme unveils a connection
of the quantum resonance (supermetallic phase) of kicked rotors to the
periodic multi-baker map~\cite{Dorfman1, Dorfman2, Dorfman}.

\subsection{General expression of $E(t)$}
\label{sec:energy_profile}

To this end we compactify the angular momentum space with the periodic
boundary condition so that it includes $M$ unit cells. Recall that the
initial condition is not essential. We consider the energy growth with
the initial state $|\psi(0)\rangle=|\tilde n\rangle$ and average the
energy profile with respect to $\tilde n$.
Mathematically, this is equivalent to preparing an ``equilibrium state'',
$\rho_{eq}=(qM)^{-1}\sum_{\tilde n} |\tilde n\rangle\langle \tilde n|$.

\begin{figure}
\includegraphics[width=8.7cm] {fig8.eps}
\caption{(Color online) The simulation results, green solid line
in (a) and black dash-dotted line in (b), of the energy growth
corresponding to the two cases in Fig.~\ref{fig:7}. They exhibit
again the universal metallic-supermetallic growth crossover as
analytically predicted by Eq.~(\ref{eq:7}) for the orthogonal
(blue dashed line) and by Eq.~(\ref{eq:8}) for the unitary (red
dotted line) symmetry.}
\label{fig:8}
\end{figure}

By the definition of $E(t)$ we find
\begin{eqnarray}\label{eq:34}
    E(t) = \frac{1}{2}\sum_{\tau_{1,2}=0}^{t-1}{\rm Tr}(\rho_{eq}
    \hat v(\tau_1) \hat v(\tau_2))
    \equiv \frac{1}{2}\sum_{\tau_{1,2}=0}^{t-1} C_{\tau_1,\tau_2},
\end{eqnarray}
where $\hat v(\tau)\equiv \frac{K}{\tilde h}\sin\hat\theta(\tau)$ may
be considered as the velocity (corresponding to the motion in angular
momentum space) operator, with
\begin{equation}\label{eq:35}
    \hat v(\tau) = (\hat U^\dagger)^\tau \hat v\, \hat U^\tau,
\end{equation}
and $C_{\tau_1,\tau_2}$ is its auto-correlation function. Equation (\ref{eq:34})
is further reduced to
\begin{eqnarray}\label{eq:36}
    E(t)= \frac{1}{2}\left(C_0 t+2\sum_{\tau=1}^{t-1} (t-\tau)C_{\tau}\right)
\end{eqnarray}
thanks to $C_{\tau_1,\tau_2}=C_{\tau_1-\tau_2}$.

Next, we calculate the auto-correlation function $C_\tau$. It can be
rewritten as
\begin{eqnarray}\label{eq:37}
    C_\tau=\left(\frac{K}{\tilde h}\right)^{2}\frac{1}{qM}{\rm Tr}
    ((\hat U^\dagger)^\tau \sin\hat \theta\hat U^\tau \sin\hat \theta)
\end{eqnarray}
by using its definition [cf. Eq.~(\ref{eq:34})]. Employing the Bloch's
theorem (\ref{eq:3}) we obtain
\begin{eqnarray}
    C_\tau &=& \left(\frac{K}{\tilde h}\right)^{2}\frac{1}{qM}\sum_{\alpha,
    \alpha'}\sum_{\theta,\theta'} e^{-i(\epsilon_\alpha(\theta)-
    \epsilon_{\alpha'}(\theta'))\tau}\nonumber\\
    &\times&|\tilde {\bf J}_{\alpha\theta,\alpha'\theta'}|^2,\label{eq:38}\\
    \tilde {\bf J}_{\alpha\theta,\alpha'\theta'} &=& \langle \psi_{\alpha,\theta}
    |\sin\hat\theta|\psi_{\alpha',\theta'}\rangle.
    \label{eq:39}
\end{eqnarray}
The matrix elements of $\tilde {\bf J}$ can be simplified to
\begin{eqnarray}
\tilde {\bf J}_{\alpha\theta,\alpha'\theta'}&=&\frac{1}{2iM}\sum_{\tilde n,\tilde n'}
\varphi_{\alpha,\theta}^*(\tilde n)\varphi_{\alpha',\theta'}(\tilde n')e^{i(\tilde n
\theta-\tilde n'\theta')}\nonumber\\
&&\times\left(\delta_{\tilde n-\tilde n',1}-\delta_{\tilde n-\tilde n',-1}\right)\nonumber\\
&=&\frac{1}{2i}\delta_{\theta\theta'}\sum_{n=0}^{q-1}(\varphi_{\alpha,\theta}^*(n)
\varphi_{\alpha',\theta}(n-1)e^{i\theta}\nonumber\\
&&-
\varphi_{\alpha,\theta}^*(n)\varphi_{\alpha',\theta}(n+1)e^{-i\theta})\nonumber\\
&\equiv& \delta_{\theta\theta'}\tilde {\bf J}_{\alpha\alpha'}(\theta).
\label{eq:40}
\end{eqnarray}
In deriving the second equality we have summed over the Bravis lattice which
enforces $\theta=\theta'$. Substituting Eq.~(\ref{eq:40}) into Eq.~(\ref{eq:38})
gives
\begin{eqnarray}
    C_\tau = \left(\frac{K}{\tilde h}\right)^{2}\frac{1}{qM}\sum_{\alpha,\alpha'}
    \sum_{\theta} e^{-i(\epsilon_\alpha(\theta)-\epsilon_{\alpha'}(\theta))\tau}
    |\tilde {\bf J}_{\alpha\alpha'}(\theta)|^2.
    \label{eq:41}
\end{eqnarray}
By setting $\tau=0$ in Eqs.~(\ref{eq:37}) and (\ref{eq:41}) we find the
following identity,
\begin{eqnarray}
    \frac{1}{Lq}\sum_{\alpha,\alpha'}\sum_{\theta} |\tilde
    {\bf J}_{\alpha\alpha'}(\theta)|^2=\frac{1}{2}.
    \label{eq:46}
\end{eqnarray}
Eqs.~(\ref{eq:36}) and (\ref{eq:41}) constitute an exact formalism
for calculating $E(t)$. Formally, they are identical to the formulas giving
the mean-squared displacement in the periodic multi-baker map.
The latter essentially is composed of an infinite number of standard baker's maps,
which are coupled to each other, with the coupling being spatially periodic~\cite{note3}.
(The details are certainly not.)

\subsection{Effects of chaotic fluctuations of Bloch bands and wave functions}
\label{sec:chaotic_fluctuation}

Equation (\ref{eq:41}) shows that the correlation is governed by the
quasienergy spectrum $\{\epsilon_\alpha(\theta)\}$ and the corresponding
eigenfunctions $\varphi_{\alpha,\theta}$, which determine the oscillatory
factor $e^{-i(\epsilon_\alpha(\theta)-\epsilon_{\alpha'}(\theta))\tau}$
and the matrix elements of the velocity operator, i.e., $\tilde
{\bf J}_{\alpha\alpha'}(\theta)$, respectively. Because the reduced
quantum system in a unit cell (for given Bloch angle $\theta$) is
chaotic, both $\{\epsilon_\alpha(\theta)\}$ and $\varphi_{\alpha,\theta}$
exhibit chaotic fluctuations. If we assume that the fluctuations of
$\{\epsilon_\alpha(\theta)\}$ and $\varphi_{\alpha,\theta}$ are
independent, then the correlation function is simplified to
\begin{eqnarray}
    C_\tau &\rightarrow& \left(\frac{K}{\tilde h}\right)^{2}\frac{1}{Lq}
    \sum_{\alpha,\alpha'}\sum_{\theta} \left\langle e^{-i(\epsilon_\alpha
    (\theta)-\epsilon_{\alpha'}(\theta))\tau}\right\rangle \nonumber\\
    &&\times \left\langle |\tilde {\bf J}_{\alpha\alpha'}(\theta)|^2
    \right\rangle,
    \label{eq:42}
\end{eqnarray}
where $\langle\cdot\rangle$ stands for the averages over a random ensemble
of energy spectrum and wave functions, respectively. These random ensembles
are well described by the RMT.

For the system without the inversion symmetry ($\beta=2$), following
Ref.~\cite{Dorfman} we use the RMT corresponding to CUE to find
\begin{eqnarray}
\left\langle e^{-i(\epsilon_\alpha(\theta)-\epsilon_{\alpha'}(\theta))
\tau}\right\rangle=\Bigg\{\begin{array}{c}
         1,\quad \tau=0; \\
         \frac{\tau-q}{q(q-1)},\quad 0<\tau<q; \\
         0. \quad \tau\geq q.
       \end{array}
\label{eq:43}
\end{eqnarray}
For the system with the inversion symmetry ($\beta=1$), we find
\begin{eqnarray}
&&\left\langle e^{-i(\epsilon_\alpha(\theta)-\epsilon_{\alpha'}(\theta))
\tau}\right\rangle\nonumber\\
&=&\Bigg\{\begin{array}{c}
         1,\, \tau=0; \\
         \frac{1}{q(q-1)}\left(-q+2\tau \left(f(\frac{q}{2}+\tau)-
         f(\frac{q}{2})\right)\right),\, 0<\tau<q; \\
         \frac{1}{q(q-1)}\left(-q+2\tau \left(f(\frac{q}{2}+\tau)-
         f(\tau-\frac{q}{2})\right)\right),\, \tau\geq q
       \end{array}
       \nonumber\\
\label{eq:44}
\end{eqnarray}
corresponding to COE, where $f(\tau)\equiv
\sum_{k=1}^\tau \frac{1}{2k-1}$. Note that these results are independent
of $\theta$.

We cannot calculate the second factor in Eq.~(\ref{eq:42}) analytically.
Instead, motivated by the similarity between Eq.~(\ref{eq:41}) and the
corresponding expression for the periodic multi-baker map, we hypothesize
that the matrix elements of the velocity operator have the universal
behavior. So, we translate the results of Ref.~\cite{Dorfman} obtained
from the RMT to the present context which read
\begin{eqnarray}
\left\langle |\tilde {\bf J}_{\alpha\alpha'}(\theta)|^2\right\rangle=
\bigg\{\begin{array}{c}
         \frac{1}{2}\frac{3-\beta}{q+3-\beta},\quad \alpha=\alpha'; \\
         \frac{1}{2}\frac{q}{(q-1)(q+3-\beta)},\quad \alpha\neq \alpha',
         \end{array}
\label{eq:45}
\end{eqnarray}
where the overall factor of $1/2$ makes it obey the identity (\ref{eq:46}).
Below key predictions obtained from this ansatz are confirmed.

We substitute Eqs.~(\ref{eq:43})-(\ref{eq:45}) into Eq.~(\ref{eq:42}).
In combination with Eq.~(\ref{eq:36}) we obtain for $\beta=1$
($D_q=(\frac{K}{2\tilde h})^2$)
\begin{eqnarray}
\frac{E(t)}{qD_q}=\bigg\{\begin{array}{c}
  \tilde t + \frac{\tilde t (\tilde t-q^{-1})}{(1+2q^{-1})}\left(1+
  \frac{\tilde t-2q^{-1}}{3(1-q^{-1})}\right),\quad 0<\tilde t<1; \\
  \frac{2\tilde t^2}{1+2q^{-1}}+\frac{1+q^{-1}}{3(1+2q^{-2})},
  \quad \tilde t\geq 1,
  \end{array}
\label{eq:48}
\end{eqnarray}
where small corrections have been ignored, and for $\beta=2$
\begin{eqnarray}
\frac{E(t)}{qD_q}=\bigg\{\begin{array}{c}
  \tilde t + \frac{\tilde t (\tilde t-q^{-1})(\tilde t-2q^{-1})}
  {3(1-q^{-2})},\quad 0<\tilde t<1; \\
  \frac{\tilde t^2}{1+q^{-1}}+\frac{1}{3}, \quad \tilde t\geq 1.
  \end{array}
\label{eq:47}
\end{eqnarray}
As shown in Fig.~\ref{fig:2}, these results are in excellent agreement
with those predicted by the field theory and are confirmed by simulations
as shown in Sec.~\ref{sec:simuresults}. Indeed, for $\beta=2$,
Eq.~(\ref{eq:47}) is identical to the analytic expression (\ref{eq:8})
for $q\gg 1$. Equations (\ref{eq:48}) and (\ref{eq:47}) show that
\begin{equation}\label{eq:49}
\frac{E|_{\beta=1}(t)}{E|_{\beta=2}(t)}\stackrel{t\gg 1}
{\longrightarrow} 2,\quad {\rm for}\, q \gg 1.
\end{equation}

Summarizing, the universal metal-supermetal dynamics crossover can
be attributed to universal chaotic fluctuations of Bloch bands and
eigenfunctions. Since the latter are well described by the RMT, we
expect that such dynamics universality is identical to the RMT
universality. On the other hand, it is well known~\cite{Altland97}
that very rich RMT universality classes are brought about by different
symmetries. Although in this work only crossovers corresponding to the
orthogonal and unitary classes are found, the richness of RMT universality
classes implies that even richer universal crossover behaviors may exist,
provided the system exhibits other symmetries.

\section{Conclusion}
\label{sec:conclusion}

We analytically and numerically studied the dynamics of a large
class of generalized kicked rotor systems the Hamiltonian of which is given by
Eq.~(\ref{eq:13}) with rational $\tilde h/(4\pi)=p/q$. The universal laws
for the crossover from metal to supermetal dynamics are found, manifesting
themselves in universal crossovers from metallic (linear) to supermetallic
(quadratic) energy growth. These crossover behaviors are determined only by
the system's symmetry and insensitive to the details such as the values of
$p$ and $q$ and the strength of the kicking potential which only affect the
energy and the time scales. Specifically, we find that when the Hamiltonian
$H_0(\hat n)$ governing the free rotation has inversion symmetry, i.e.,
$H_0(\hat n)=H_0(-\hat n)$, the energy profile follows an orthogonal-type
crossover behavior, described by Eq.~(\ref{eq:7}), and otherwise the
crossover is a unitary type, described by Eq.~(\ref{eq:8}). We show by the
RMT that these universal dynamics crossovers can be attributed to the
universal Bloch wave functions and band fluctuations of quantum systems
reduced to a unit cell. This leads us to conjecture that in more general
chaotic systems with periodic driving, the universality class of the
metal-supermetal dynamics crossover is identical to that of eigenfunction
and spectral fluctuations described by the RMT.

Our results can be generalized to condensed-matter systems (such as
semiconductors) where Bloch bands have been seen to exhibit chaotic
fluctuations following the RMT. For these systems even richer universality
classes have been predicted~\cite{Altland97}. The ensuing metal-supermetal
dynamics crossovers are expected to fall into different universality classes
manifesting in different universal behaviors of the optical conductivity.
Provided the unit cell is so complicated that the Bloch energy spectrum
exhibits chaotic fluctuations following the Wigner-Dyson statistics, our
results are expected to be valid.

Finally, we remark that in Ref.~\cite{Guarneri09} the supermetallic growth
has been rigorously established for the standard kicked rotor under general
parametric conditions. This result was obtained without resorting to the
chaoticity condition. A byproduct of the present work, i.e., $E(t\rightarrow
\infty) \sim t^2$, is consistent with this rigorous result, but is established
based on the chaoticity condition. We believe that our main result of universal
metal-supermetal dynamics crossovers is of chaos origin. Such a crossover
phenomenon was not studied in Ref.~\cite{Guarneri09}.

\section*{Acknowledgements}
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11174174, 11275159, 11335006,
and 11535011).

\begin{appendix}
\section{\label{sec:derivation}Derivations of Eq.~(\ref{eq:7})}

In this Appendix we derive Eq.~(\ref{eq:7}). To this end we employ the
polar coordinate representation of $Q$ \cite{Efetov97}. Specifically, we parametrize $Q$ as
\begin{eqnarray}
Q=RQ_0R^{-1},\quad Q_0=\left(
\begin{array}{rcl}
\cos\hat{\Theta} &i\sin\hat{\Theta}\\
-i\sin\hat{\Theta} &-\cos\hat{\Theta}
\end{array}
\right)_{\rm AR},
\label{eq:28}
\end{eqnarray}
where $Q_0$ is the radial part and
\begin{eqnarray}
\hat{\Theta}&=&\left(
\begin{array}{rcl}
\hat{\theta}_{11} &0\\
0 &\hat{\theta}_{22}
\end{array}
\right)_{\rm BF},\label{eq:29}\\
\hat{\theta}_{11}&=&\left(
\begin{array}{rcl}
\tilde{\theta} &0\\
0 &\tilde{\theta}
\end{array}
\right)_{\rm T},\quad
\hat{\theta}_{22}=i\left(
\begin{array}{rcl}
\tilde{\theta}_1 &\tilde{\theta}_2\\
\tilde{\theta}_2 &\tilde{\theta}_1
\end{array}
\right)_{\rm T}
\nonumber
\end{eqnarray}
with ${0<\tilde{\theta}<\pi,\tilde{\theta}_{1,2}>0}$, and $R$ is the
transverse part commuting with $\sigma_{\rm AR}^3$. It is important to
note that the Grassmann variables enter only into $R$ (see
Ref.~\cite{Efetov97} for details).

We substitute this parametrization into Eqs.~(\ref{eq:17}) and (\ref{eq:27}).
Taking into account that Eq.~(\ref{eq:19}) now is simplified to
\begin{equation}\label{eq:32}
    \hat \vartheta=\left(
                     \begin{array}{cc}
                       \theta_+ & 0 \\
                       0 & \theta_- \\
                     \end{array}
       \right)_{\rm AR}\otimes \sigma_{\rm T}^{0}\otimes \sigma_{\rm FB}^0,
\end{equation}
we find
\begin{eqnarray}
K_\omega&=&\frac{1}{4} \int_1^\infty d\lambda_1 \int_1^\infty d\lambda_2
\int_{-1}^1 d\lambda \, e^{-S} \nonumber\\
&\times&\frac{(1-\lambda^2)(1-\lambda^2-\lambda_1^2-\lambda_2^2+2\lambda_1^2
\lambda_2^2)} {(\lambda^2+\lambda_1^2+\lambda_2^2-2\lambda \lambda_1 \lambda_2-1)^2},
\label{eq:30}
\end{eqnarray}
where the radial coordinates ${\lambda\equiv \cos\tilde{\theta}, \lambda_{1,2}
\equiv \cosh\tilde{\theta}_{1,2}}$ and the zero-mode action
\begin{eqnarray}
S&=&\frac{\pi}{2\Delta}[D_q(\Delta\theta)^2(2\lambda_1^2\lambda_2^2-\lambda^2-
\lambda_1^2-\lambda_2^2+1)\nonumber\\
&&\quad\quad + 2i\omega(\lambda-\lambda_1\lambda_2)].
\label{eq:31}
\end{eqnarray}
Substituting Eq.~(\ref{eq:30}) into Eq.~(\ref{eq:16}) we obtain Eq.~(\ref{eq:7}).

\end{appendix}


Welcome to the new and improved Fiddaman Australia website.
In 2003, even before it became fiddaman.info, I started the original website using Microsoft FrontPage. In 2018, the site was reconstructed using WordPress, although some components of the old site still remain. To complete the picture, I have also included some valuable family and personal history pictures prior to 2003.
The header on every page is randomly select from a large number of my pictures that were suitable for trimming into the banner shape.
The Recent Posts area (above, to your right) shows only the most recent 5 months of news and pictures. While WordPress lists them in reverse order, most recent at the top, the content of each post is in chronological order, with links at the bottom of each month to the previous and next months, so that you can easily jump to the next one and continue the flow.
The Previous Months (also to your right) are a full record of the material here. Because the archive is a collection of everything published in that month (which on this site is just the one post), you don't see the links to previous and next. However, you can get these and follow the flow by clicking on the shortened TITLE of the post up near the top of the page, just under the "month", which takes you to the POST for that month. Clear as mud, right?
There are a few other things on the right that are either really cool, or useful to me or to the people who search for them.
I would venture to suggest that I have used it well.


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I'm lucky this time because I doubt another person would refuse a lower price.
I'm still the cheapest on the block in order to make my services accessible to everyone, but it's not free anymore and I'm having a hard time charging money.
I wasn't like this before. I used to be able to ask for raises and sell the crap out of myself. I was a confident business woman, who could play hardball with the best of them. But, out here on my own, it's totally up to me to make it happen, and I feel grasping and unworthy.
Like Oliver Twist, please sir, may I have some more.
I think a lot of people feel this way, especially if they are young and trying to forge their career path and it's a shame because I'm sure a lot of us aren't being rewarded for what we actually do.
I don't think that nervousness in the pit of my stomach will go away soon, but I think I finally figured out how to deal.
I quoted another client this week. I reminded myself that clients are just people. They are not some all-knowing-all seeing god of my monetary universe. Just people who are resourceful and smart and if they can't pay the price, they will say so. Then, the ball is in my court to come down in price or try knocking on another door.
I'm happy to report that so far, every quote has turned into a new client, which means my prices are fair and reasonable. YAY!! I'm feeling good about 2012.
So please learn from my craziness. Whether you're an entrepreneur charging a client or an employee asking for a raise, you have to ASK and hold your ground.
If you are an entrepreneur, remember that the client came to you in the first place, so you clearly have something they want. Value the service or good you are providing.
As an employee, when you head into your review, remember that the people on the other side of the desk are just people, and they are EXPECTING you to ask for a raise, that's what reviews are all about. So, you may as well do it.
The worst thing that can happen is that they say no. At least you will know where you stand.
We all have value and have a right to be recognized for it, so get out there and play hard ball.


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Q: Mootools 1.3.1: Selector for parent (id) > child I am currently working on a mootools 1.3.1 script and have a structure like this:

I know already the selector $$('div.class img') but how does it look for an id?
The whole documentation for css selectors on mootools.net is missing: http://mootools.net/docs/core/Utilities/Selectors
And here is only for class?: http://mootools.net/docs/core/Slick/Slick

A: Just replace the class selector with an ID selector?
$$('div#100 img')

If you only want img elements that are children of div#100, add a child combinator as in your question title:
$$('div#100 > img')

If MooTools says it supports CSS selectors, it shouldn't be that much of a difference from the standard CSS that we use. If the latest documentation is somehow missing, it should still be sufficient to fall back to a previous but still recent version (here's the documentation for MooTools 1.2.5, the second Google result for "mootools selectors") or regular CSS documentation, like the W3C selectors spec.

redpajama_set_name: RedPajamaStackExchange


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I have owned this Formula for the past 13 years, and have updated, or replaced to many items to mention. I have a complete mechanical history and receipts f as a "WordDoc" that I can email to anyone interested. I've driven the car an average of 2000 miles a year since ownership. I have a service manual, the Enthusiasts Guide to Pontiac Fiero's, a Haynes Repair Manual and my three ring binder of history. I also have the four original factory 15" "Black Diamond Spoke wheels.
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This quote might have been new to some of us at the beginning of the week. But we've been acting as the Body of Christ our whole lives, whether we knew it or not. I feel safe to say that each person here regularly uses their eyes, hands, feet, and body to proclaim the Good News of God's Love.
I wonder if some of you might be willing to share ways in which you act as the body of Christ every day, or every week, even every once in a while. How are you the eyes, hands, feet, or body of Christ? I am sure we could sit here all day and list out ways in which we continually seek to serve Christ in one another. Here's the challenge: in every moment and in every choice, we make there is an opportunity for us to be Christ's presence in the world. It is up to us to say yes to the call, to choose to act out of love. This love is part of our nature because we were made in the image of the God of love.
When we accept this as truth it changes our lives. When we do everything out of love – beautiful things happen. Our relationships are healed and restored, they become whole. Our communities are healed and restored, they become whole. If we all acted out of love all of the time, the whole world would be healed and restore, all of creation would be whole.
In our Gospel this morning we hear the great commandments: "Love the Lord your God with all your soul and with all your mind." And: "Love your neighbor as yourself." From this we learn the most important thing we can do is love. We are called to do so freely and unconditionally.
We don't have to be all things to all people. You don't have to try to be the foot and the hand and the eye and the ear all of the time. We all have God given gifts. Sometimes we are called to be the eyes to notice where the world around us needs healing. Sometimes we are the feet and are called to go to where our gifts can be put to use. Sometimes we are a listening ear or a helping hand or a mouth speaking the truth in love. Yet, this is always true: "Christ has no body now on earth but yours." You will always be the body, so go forth and live in love.


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Home / FAQs / More licenses are being used than predicted. What do I do?
You're actually out of licenses. For example: you have a license for 100 devices, all of them active and used, and the Agent is installed on every one of them.
You frequently reinstall/reimage devices and old devices don't get cleared of licenses.
In this case please contact us to upgrade your number of licenses. The Manager will notify you about this between 80 and 100% of licenses used.
If the ESD Agent section in the Manager is not expanded, double click to expand it.
Right-click on Report Directory and click Verify Computers. The Manager will now check against Active Directory and verify if the devices exist.
If there are devices found in the Report share which are not in Active Directory you will get the above dialog. Click Yes to clear them from the Report share and subsequently clear the licenses.
The above dialog will show if all devices found in the Report share are also found in Active Directory.


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Talking About Books
~ A space to share thoughts on books, reading and writing
Warlight: Michael Ondaatje
Posted by suroor alikhan in Fiction
Review by Usha Raman
"The lost sequence in a life, they say, is the thing we always search out."
"We order our lives with barely held stories"
Memory is a strange thing; it reveals the ephemeral nature of experience and the power of its reconstruction. It builds stories where none existed. And it imbues monochromatic lives with colours drawn more from imagination than life. But we need memory in order to make meaning of life, and in Michael Ondaatje's Warlight, Nathaniel's search for the lost sequence of his mother's life is in many ways the search for the meaning of his own. The novel is set in the confusing time after the second world war, when London is still limping back to a normalcy that will forever remain elusive, and nothing is what it seems—and neither, as Nathaniel discovers, are people.
Abandoned by their parents to the guardianship of a mysterious and unknowable man they fondly dub The Moth, Nathaniel and his sister Rachel spend their teenage years in an adventurous fog, encountering a series of not-quite-mainstream adults, none of whom seem to be engaged in easily explainable occupations. There is the Darter, a one-time boxer and greyhound-smuggling, gun-running womanizer who, unmindful of propriety or danger, takes the young Nathaniel on his nighttime river runs and introduces him to the women in his life. Among these is Olive Lawrence, a cloud-reading ethnographer who has a surprising connection with the intelligence services.
The children's discovery of duplicity—or intrigue—begins when they find their mother's packed trunk left behind in the basement, revealing her departure, ostensibly to join their father in his posting to Singapore, to be a charade. While Rachel takes this to be a betrayal that never be forgiven, even when her mother returns to reclaim her life with them, Nathaniel seeks to fill out the years of her absence, to fill in the shadowy outlines of a past that could, possibly, give him a sense of the person she was. And in that process, find that his own memories are fickle things, meaning one thing at one moment, rendered false the next.
It's no coincidence that as an adult Nathaniel finds himself recruited to work in the archives of war intelligence, a fitting place in which to recover the memory of his mother, a key figure in wartime espionage, and her association with the oddly named Marsh Felon, the man who drew her into the ring of spies and away from her life in domesticity. The fragments of their story unfold in the dim light left behind by war records, buried deep in the filing cabinets of the British secret service. And in the pursuit of his mother's life, he is forced to rearrange the meanings of his own memories—sometimes with devastating results.
Warlight has been my introduction to Ondaatje, a writer whom I have been meaning to read for years. It has left me thirsting for more of the textured prose, the spare yet powerful characterization, and the suggestive outline of plot that seem to be his forte. You want to pause and drink in the meanings that seem to speak not just to and from the characters in the book, but to your life as well. This is his eighth novel, one that The Guardian hails as another instance of "magic from a past master". Clearly, I have a lot of magic to catch up on.
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A Haunted House and Other Stories: Virginia Woolf
Go, Went, Gone: Jenny Erpenbeck
The Places in Between: Rory Stewart
Magpie Murders: Anthony Horowitz
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Dr. Gary Whetstone's Bio Pastor Faye Whetstone's Bio Magazine article Location and Directions Service Times What We Believe History Publishing Requests Press Release Pastor Gary's Itinerary Privacy Statement
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Working Closer to Home
GWWM and Victory Christian Fellowship is involved on both Domestic and Urban missions. The great comission of activity that Jesus Christ has commanded has no boundaries and the mission field is world.
Blessings Dressings and More - Every Saturday the Blessings Dressings and more ministry reaches out to the community providing food and cloting to those in need.
The Neighborhood Evangelism Team (NET) - Thousands of people have been reached with the Gospel of Jesus Christ through the work of the GWWM Neighborhood Evangelism Team. The NET mees every Saturday at 9:30am for prayer and then a t 10:00am they take to the streets to be a witness for Jesus Christ.
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