# B Unitarity and thermodynamics?

1. Mar 2, 2016

### Collin237

A paper I'm reading says that the 2nd Law of Thermodynamics is related to unitarity. And it references:

S. Weinberg, "The Quantum theory of fields. Vol. 1: Foundations"

Does anyone here know what this might mean?

2. Mar 2, 2016

### vanhees71

I don't know, where in Weinberg's marvelous book you find a comment on the thermodynamical arrow of time, but the relation between the 2nd law and unitarity is indeed very important.

You derive the Boltzmann transport equation either from classical mechanics or quantum theory by making a subtle approximation which seems innocent but is at the heart of getting a "time arrow", i.e., a preference in the direction of time, which we experience in everyday life. E.g., if you push something and let it go it will come to a stop through friction, i.e., the kinetic energy is dissipated away from the body, heating it and the environment up by the corresponding amount of energy, i.e., energy is dissipated to a lot of degrees of freedom and the ordered motion comes to a stop and finally the object comes to equilibrium with its surrounding. Supposed you deal with microscopical laws that are time-reversal invariant the opposite behavior does not violate any physical law, i.e., in principle it could happen that an object sitting somewhere spontaneously takes heat energy and spontaneously accelerates in some direction, using this energy. However, you'll never observe such an oddity, because it is very very very... unlikely that the random motion of the molecules in the environment of the object is all of a sudden conspiring to push it in the certain direction. And so it practically never ever happens.

To derive this behavior from physical laws that are time-reversal invariant (and that's the case for everything in nature except the feeble weak interaction) you have to bring in this "arrow of time" by some approximation. I'll argue quantum mechanically. You have a large system of molecules (order $10^{24}$ particles) in a container. So you cannot describe such a system in detail, but experience shows that you can describe such a box of a gas with help of thermodynamics, i.e., some relevant macroscopic parameters. One level of description is to look at the one-particle phase-space distribution function, i.e., you count the number of particles in some region $\Delta^3 \vec{x}$ in space and $\Delta^3 \vec{p}$ in momentum space (which is together phase space). This phase-space element should be of a size that it still contains a lot of particles but small compared to the scales upon which the mean position and momentum of such a fluid-phase-space cell varies typically.

Now the total change of the number of particles in such a phase-space cell per phase-space volume with time is
$$D_t f(t,\vec{x},\vec{p})=\partial_t f(t,\vec{x},\vec{p})+\vec{p}/m \cdot \vec{\nabla}_x f(t,\vec{x},\vec{p}).$$
But how can such a change come about from a microscopic point of view? It comes from collisions of the molecules which each other. Now we also assume that the gas is pretty dilute in the following sense: We assume that for quite a long distance each molecule just moves freely (large mean-free path) and only rarely two particles collide elastically with each other, and the duration of this collision process is much shorter than the time during which the particles travel freely. Then we can use the usual quantum-mechanical scattering prescription, i.e., two asymptotic free particles approach each other, collide and then fly appart relaxing again to an asymptotic free state (note that here we assume that the collisions are through a short-range interaction, i.e., the potential falls faster than with 1/distance, i.e., it does not apply to Coulomb forces of charged particles!).

Now comes the crucial point concerning the question of the arrow of time! We want to stay with our description on the level of the single-particle phase-space distribution. For our collision we need the distribution of particle pairs (!) in the initial state, and we assume that this two-particle phase-space distribution is given by the product of the single-particle distributions! That means that we ignore any possible statistical correlations between any two particles. Formally this is called the truncation of the socalled BBKYZ hierarchy (for details, see Landau-Lifshitz vol. 10). The assumption is also known as Boltzmann's hypothesis of molecular chaos. Then the change in the phase-space distribution function is given by the two-particle elastic-scattering amplitude in terms of the Boltzmann collision integral (I neglect for simplicity Bose or Fermi statistics, which would lead to Bose enhancement and Pauli blocking of the final states of the collision and lead to the socalled Boltzman-Uehling-Uhlenbeck equation, which doesn't change anything principle in our considerations concerning the arrow of time).

$$C[f]=\frac{1}{2} \int \mathrm{d}^3 \vec{p}_2 \mathrm{d}^3 \vec{p}_3 \mathrm{d}^3 \vec{p}_4 [|M(\vec{p}_3,\vec{p}_4 \rightarrow \vec{p},\vec{p}_2]|^2 f(t,\vec{x},\vec{p}_3) f(t,\vec{x},\vec{p}_4) - |M(\vec{p},\vec{p}_2 \rightarrow \vec{p}_3,\vec{p}_4]|^2 f(t,\vec{x},\vec{p}) f(t,\vec{x},\vec{p}_2) \delta(E+E_2-E_3-E_4) \delta^{(3)}(\vec{p}+\vec{p}_2-\vec{p}_3-\vec{p}_4).$$
The $\delta$ distributions take into account energy-momentum conservation, and I have minimized the amount of constant factors lumping them all into the scattering-matrix elements squared.

Now comes the unitarity argument. From the unitarity of the S-matrix you can conclude that in the above formula you can set the two matrix elements squared equal, i.e., you can write the collision term as
$$C[f]=\frac{1}{2} \int \mathrm{d}^3 \vec{p}_2 \mathrm{d}^3 \vec{p}_3 \mathrm{d}^3 \vec{p}_4 |M(\vec{p}_3,\vec{p}_4 \rightarrow \vec{p},\vec{p}_2]|^2 [f(t,\vec{x},\vec{p}_3) f(t,\vec{x},\vec{p}_4) - f(t,\vec{x},\vec{p}) f(t,\vec{x},\vec{p}_2)] \delta(E+E_2-E_3-E_4) \delta^{(3)}(\vec{p}+\vec{p}_2-\vec{p}_3-\vec{p}_4).$$
This is known as the (weak) principle of detailed balance and is crucial for the Boltzmann H-theorem to be proofable from this equation. You just have to define entropy and show with help of the Boltzmann equation that it never decreases. The proof is lengthy but not too difficult. For the details see Landau-Lifshitz vol. X or my lecture notes (which deal with the relativistic Boltzmann equation, but that's a minor technical detail; the basic principles concerning the arrow of time and the H-theorem are exactly the same):

http://fias.uni-frankfurt.de/~hees/publ/kolkata.pdf

3. Mar 3, 2016

### A. Neumaier

The paper refers to the end of Section 3.6 of Weinberg's book, where he derives the Boltzmann H theorem (3.6.20) from the optical theorem (3.6.10). In his derivation he introduces the arrow of time when he writes (without discussion) equation (3.6.19). This is not valid without making some implicit assumption (that turns the reversible unitary dynamics into a dissipatice one) of the kind spelled out in vanhees71's post:

Last edited: Mar 3, 2016
4. Mar 3, 2016

5. Mar 5, 2016

### Collin237

I take it that p and p2 are incoming momenta, and p3 and p4 are outgoing momenta? And I take it that the norm of a matrix X is Tr(X^T X)?

So there are two matrices that should have the same norm if unitarity is exact. You substitute one norm for the other and break the time symmetry. I take it this is assumed to in principle give a slightly different value for the collision term (or else the substitution would be a mere symbolic simplification), so as to implicitly deny the assertion of exact unitarity without having to go out on a limb guessing what the non-unitary component might be?

6. Mar 5, 2016

### vanhees71

$p$ and $p_2$ are incoming momenta for the loss term and outgoing ones for the gain term in the Boltzmann collision integral. For a discussion of unitarity and detailed balance, please see p. 27 of

http://fias.uni-frankfurt.de/~hees/publ/kolkata.pdf

7. Mar 5, 2016

### Collin237

@Demystifier: I thought of that too, but I wasn't going to mention it. It would sound too Prigogine.

8. Mar 5, 2016

### Demystifier

Thought of what?