<p>The current model (<span class="math-container">$\Lambda$</span>CDM) predicts that space will expand exponentially, and keep on doing it indefinitely far into the future. The end result, after the stars burn out, galaxies disperse, protons decay and black holes evaporate is a very thin soup of stable particles (see <a href="https://arxiv.org/abs/astro-ph/9701131" rel="nofollow noreferrer">Adams &amp; Laughlin 1997</a> or their popular book <em>The Five Ages of the Universe</em>). </p>

<p>Since each stable particle finds itself separated by exponentially growing distances due to the expanding space they will not be able to interact. Their density scales as <span class="math-container">$\rho(t)\sim \rho_0/a(t)^3 = \rho_0 e^{-3t/H_0}$</span>. The probability of a collision over time <span class="math-container">$dt$</span> for a particle is roughly <span class="math-container">$\sigma v \rho(t) dt$</span>, which if we integrate to infinity becomes <span class="math-container">$\Pr[\text{any  collision}]=\sigma v \rho_0 \int_0^\infty e^{-3t/H_0} dt = \sigma v \rho_0 H_0/3 $</span> which is finite - there will only be a finite number of interactions for each particle, ever.</p>

<p>At this point temperatures are extremely low since all background radiation has been redshifted to undetectable levels, and only the very cold horizon radiation remains (<span class="math-container">$\approx 10^{-30}$</span> K). Particles end up in a equilibrium state with this heat bath, and this is the real heat death of the universe.</p>

<p>This low temperature is still enough to rarely excite the particles, or generate new particles... <a href="https://physics.stackexchange.com/questions/434877/can-the-universe-build-itself-back-together-after-the-heat-death/434886#434886">or Boltzmann brains or entire worlds</a>. Maybe. Eternity is a very long time, so low-probability processes will happen if they are allowed. They will however be so rare that the effects of one will typically dissipate <em>long</em> before anything else happens.</p>

CC BY-SA 4.0
2019-05-03T22:20:47.707Z
Presuming protons *do* decay (I'm sceptical).
PM 2Ring
CC BY-SA 4.0
2019-05-03T23:28:14.970Z
@PM2Ring - There are quite a lot of possible baryon-nonconserving mechanisms theorized. But even if there are none, over very long spans free energy minimization might cause dissolution of matter since the entropy gain can be large; see the discussion in http://math.ucr.edu/home/baez/end.html
Anders Sandberg
CC BY-SA 4.0
2019-05-03T23:43:52.497Z
Thanks, I'll read it shortly. I'm familiar with https://en.wikipedia.org/wiki/Timeline_of_the_far_future and have often linked it, but I love the way John Baez writes.
PM 2Ring
CC BY-SA 4.0
2019-05-03T21:59:39.350Z
2019-05-03T21:59:39.350Z
Anders Sandberg
<p>.. because, how would all the current matter and energy manifest then? Isn't everything coming to complete stop, minimal temperature, zero interaction state impossible? Wouldn't there still be some rare minor events occuring? And if so, given infinite amount of time couldn't these result in events resulting in some kind of chain-reaction?</p>

CC BY-SA 4.0
2019-05-03T11:53:02.220Z
AFAIK, the total energy of the universe is zero, because it also combines potential energies of various interactions, so it should violate this law.
Tosic
CC BY-SA 4.0
2019-05-03T12:14:46.230Z
The expansion of space is relevant here -- the space between particles could be growing fast enough that they don't meet.
Steve Linton
CC BY-SA 4.0
2019-05-03T12:53:21.590Z
@Tosic IS zero-energy universe an accepted theory? I haven't heard of gravity being counted as negative energy, as it suggests.
marko-36
CC BY-SA 4.0
2019-05-03T12:57:29.187Z
@SteveLinton, could the universe be expanding so fast to prevent even occasional neutrinos or photons to collide? Even if it was expanding at the speed of light or way faster, this speed would have little effect within the distances between particles inside the universe.. I think.
marko-36
CC BY-SA 4.0
2019-05-03T13:41:51.840Z
Energy conservation is not a law that applies to the evolution of the universe as a whole.
ProfRob
CC BY-SA 4.0
2019-05-03T13:44:11.017Z
See https://astronomy.stackexchange.com/questions/1498/expansion-again-where-does-the-energy-come-from/1500#1500 https://astronomy.stackexchange.com/questions/26883/what-explains-the-existence-of-energy-matter-if-it-cannot-be-created-or-destroye
ProfRob
CC BY-SA 4.0
2019-05-03T15:42:50.960Z
It shouldn't be, no, actually, I made a mistake there (I even confused the Big Freeze with the Big Crunch)... I'm sure the provided links will help you much more than I can.
Tosic
CC BY-SA 4.0
2019-05-03T23:37:57.817Z
It's not that the energy is diminished. It's that everything gets more and more diluted as the universe expands. Eventually all the universe will basically become a whole lot of nothing, mostly empty space.
Florin Andrei
CC BY-SA 4.0
2019-05-04T06:12:54.900Z
It is a bit tricky to speak of an Energy balance. Dark energy is in tge the standard model but no corresponding potential energy is (currently) there.  Thermal death is reached asymptotically. What goes to zero asymptotically is energy density.
Alchimista
CC BY-SA 4.0
2021-07-29T19:24:57.453Z
This question should be superseded by the more overarching question: "Is the law of energy conservation valid for the universe as a whole, when it is considered as an isolated system?"
Alex
<fate-of-universe>
Doesn't the energy conservation law imply, that the universe can't come to a Big Freeze?
marko-36
# Doesn't the energy conservation law imply, that the universe can't come to a Big Freeze?

(Asked by: username_0 on 2019-05-03)

.. because, how would all the current matter and energy manifest then? Isn't everything coming to complete stop, minimal temperature, zero interaction state impossible? Wouldn't there still be some rare minor events occuring? And if so, given infinite amount of time couldn't these result in events resulting in some kind of chain-reaction?

**Tags:** fate-of-universe
> **Comments:**
> **username_1**: AFAIK, the total energy of the universe is zero, because it also combines potential energies of various interactions, so it should violate this law.  
> **username_2**: The expansion of space is relevant here -- the space between particles could be growing fast enough that they don't meet.  
> **username_0**: @Tosic IS zero-energy universe an accepted theory? I haven't heard of gravity being counted as negative energy, as it suggests.  
> **username_0**: @SteveLinton, could the universe be expanding so fast to prevent even occasional neutrinos or photons to collide? Even if it was expanding at the speed of light or way faster, this speed would have little effect within the distances between particles inside the universe.. I think.  
> **username_3**: Energy conservation is not a law that applies to the evolution of the universe as a whole.  
> **username_3**: See https://astronomy.stackexchange.com/questions/1498/expansion-again-where-does-the-energy-come-from/1500#1500 https://astronomy.stackexchange.com/questions/26883/what-explains-the-existence-of-energy-matter-if-it-cannot-be-created-or-destroye  
> **username_1**: It shouldn't be, no, actually, I made a mistake there (I even confused the Big Freeze with the Big Crunch)... I'm sure the provided links will help you much more than I can.  
> **username_4**: It's not that the energy is diminished. It's that everything gets more and more diluted as the universe expands. Eventually all the universe will basically become a whole lot of nothing, mostly empty space.  
> **username_5**: It is a bit tricky to speak of an Energy balance. Dark energy is in tge the standard model but no corresponding potential energy is (currently) there.  Thermal death is reached asymptotically. What goes to zero asymptotically is energy density.  
> **username_6**: This question should be superseded by the more overarching question: "Is the law of energy conservation valid for the universe as a whole, when it is considered as an isolated system?"  

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## 1 answers

### Answer by username_7 on 2019-05-03. Score: 2

The current model ($\Lambda$CDM) predicts that space will expand exponentially, and keep on doing it indefinitely far into the future. The end result, after the stars burn out, galaxies disperse, protons decay and black holes evaporate is a very thin soup of stable particles (see [Adams & Laughlin 1997](https://arxiv.org/abs/astro-ph/9701131) or their popular book *The Five Ages of the Universe*).

Since each stable particle finds itself separated by exponentially growing distances due to the expanding space they will not be able to interact. Their density scales as $\rho(t)\sim \rho\_0/a(t)^3 = \rho\_0 e^{-3t/H\_0}$. The probability of a collision over time $dt$ for a particle is roughly $\sigma v \rho(t) dt$, which if we integrate to infinity becomes $\Pr[\text{any collision}]=\sigma v \rho\_0 \int\_0^\infty e^{-3t/H\_0} dt = \sigma v \rho\_0 H\_0/3 $ which is finite - there will only be a finite number of interactions for each particle, ever.

At this point temperatures are extremely low since all background radiation has been redshifted to undetectable levels, and only the very cold horizon radiation remains ($\approx 10^{-30}$ K). Particles end up in a equilibrium state with this heat bath, and this is the real heat death of the universe.

This low temperature is still enough to rarely excite the particles, or generate new particles... [or Boltzmann brains or entire worlds](https://physics.stackexchange.com/questions/434877/can-the-universe-build-itself-back-together-after-the-heat-death/434886#434886). Maybe. Eternity is a very long time, so low-probability processes will happen if they are allowed. They will however be so rare that the effects of one will typically dissipate *long* before anything else happens.

> **Comments:**
> **username_8**: Presuming protons *do* decay (I'm sceptical).  
> **username_7**: @PM2Ring - There are quite a lot of possible baryon-nonconserving mechanisms theorized. But even if there are none, over very long spans free energy minimization might cause dissolution of matter since the entropy gain can be large; see the discussion in http://math.ucr.edu/home/baez/end.html  
> **username_8**: Thanks, I'll read it shortly. I'm familiar with https://en.wikipedia.org/wiki/Timeline_of_the_far_future and have often linked it, but I love the way John Baez writes.  

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