H-theorem

Soon after Boltzmann published his H theorem, Johann Josef Loschmidt objected that it should not be possible to deduce an irreversible process from time-symmetric dynamics and a time-symmetric formalism. If the H decreases over time in one state, then there must be a matching reversed state where H increases over time (Loschmidt's paradox). The explanation is that Boltzmann's equation is based on the assumption of "molecular chaos", i.e., that it follows from, or at least is consistent with, the underlying kinetic model that the particles be considered independent and uncorrelated.[13] It turns out that this assumption breaks time reversal symmetry in a subtle sense, and therefore begs the question. Once the particles are allowed to collide, their velocity directions and positions in fact do become correlated (however, these correlations are encoded in an extremely complex manner).[13] This shows that an (ongoing) assumption of independence is not consistent with the underlying particle model.

Boltzmann's reply to Loschmidt was to concede the possibility of these states, but noting that these sorts of states were so rare and unusual as to be impossible in practice. Boltzmann would go on to sharpen this notion of the "rarity" of states, resulting in his famous equation, his entropy formula of 1877 (see Boltzmann's entropy formula).

As a demonstration of Loschmidt's paradox, a famous modern counter example (not to Boltzmann's original gas-related H-theorem, but to a closely related analogue) is the phenomenon of spin echo.[16] In the spin echo effect, it is physically possible to induce time reversal in an interacting system of spins.

An analogue to Boltzmann's H for the spin system can be defined in terms of the distribution of spin states in the system. In the experiment, the spin system is initially perturbed into a non-equilibrium state (high H), and, as predicted by the H theorem the quantity H soon decreases to the equilibrium value. At some point, a carefully constructed electromagnetic pulse is applied that reverses the motions of all the spins. The spins then undo the time evolution from before the pulse, and after some time the H actually increases away from equilibrium (once the evolution has completely unwound, the H decreases once again to the minimum value). In some sense, the time reversed states noted by Loschmidt turned out to be not completely impractical.

In 1896, Ernst Zermelo noted a further problem with the H theorem, which was that if the system's H is at any time not a minimum, then by Poincaré recurrence, the non-minimal H must recur (though after some extremely long time). Boltzmann admitted that these recurring rises in H technically would occur, but pointed out that, over long times, the system spends only a tiny fraction of its time in one of these recurring states.

The second law of thermodynamics states that the entropy of an isolated system always increases to a maximum equilibrium value. This is strictly true only in the thermodynamic limit of an infinite number of particles. For a finite number of particles, there will always be entropy fluctuations. For example, in the fixed volume of the isolated system, the maximum entropy is obtained when half the particles are in one half of the volume, half in the other, but sometimes there will be temporarily a few more particles on one side than the other, and this will constitute a very small reduction in entropy. These entropy fluctuations are such that the longer one waits, the larger an entropy fluctuation one will probably see during that time, and the time one must wait for a given entropy fluctuation is always finite, even for a fluctuation to its minimum possible value. For example, one might have an extremely low entropy condition of all particles being in one half of the container. The gas will quickly attain its equilibrium value of entropy, but given enough time, this same situation will happen again. For practical systems, e.g. a gas in a 1-liter container at room temperature and atmospheric pressure, this time is truly enormous, many multiples of the age of the universe, and, practically speaking, one can ignore the possibility.

Since H is a mechanically defined variable that is not conserved, then like any other such variable (pressure, etc.) it will show thermal fluctuations. This means that H regularly shows spontaneous increases from the minimum value. Technically this is not an exception to the H theorem, since the H theorem was only intended to apply for a gas with a very large number of particles. These fluctuations are only perceptible when the system is small and the time interval over which it is observed is not enormously large.

If H is interpreted as entropy as Boltzmann intended, then this can be seen as a manifestation of the fluctuation theorem.

We let ${\displaystyle q_{i}}$ and ${\displaystyle p_{i}}$ be our generalized coordinates for a set of ${\displaystyle r}$ particles. Then we consider a function ${\displaystyle f}$ that returns the probability density of particles, over the states in phase space. Note how this can be multiplied by a small region in phase space, denoted by ${\displaystyle \delta q_{1}...\delta p_{r}}$, to yield the (average) expected number of particles in that region.

${\displaystyle \delta n=f(q_{1}...p_{r},t)\,\delta q_{1}\delta p_{1}...\delta q_{r}\delta p_{r}.\,}$

Tolman offers the following equations for the definition of the quantity H in Boltzmann's original H theorem.

${\displaystyle H=\sum _{i}f_{i}\ln f_{i}\,\delta q_{1}\cdots \delta p_{r}}$[18]

Here we sum over the regions into which phase space is divided, indexed by ${\displaystyle i}$. And in the limit for an infinitesimal phase space volume ${\displaystyle \delta q_{i}\rightarrow 0,\delta p_{i}\rightarrow 0\;\forall \,i}$, we can write the sum as an integral.

${\displaystyle H=\int \cdots \int f\ln f\,dq_{1}\cdots dp_{r}}$[19]

H can also be written in terms of the number of molecules present in each of the cells.

${\displaystyle {\begin{aligned}H&=\sum (n_{i}\ln n_{i}-n_{i}\ln \delta v_{\gamma })\\&=\sum n_{i}\ln n_{i}+{\text{constant}}\end{aligned}}}$[20]

An additional way to calculate the quantity H is:

${\displaystyle H=-\ln P+{\text{constant}}\,}$[21]

where P is the probability of finding a system chosen at random from the specified microcanonical ensemble. It can finally be written as:

${\displaystyle H=-\ln G+{\text{constant}}\,}$[22]

where G is the number of classical states.

The quantity H can also be defined as the integral over velocity space :

${\displaystyle \displaystyle H\ {\stackrel {\mathrm {def} }{=}}\ \int {P({\ln P})\,d^{3}v}=\left\langle \ln P\right\rangle }$

(1)

where P(v) is the probability distribution.

Using the Boltzmann equation one can prove that H can only decrease.

For a system of N statistically independent particles, H is related to the thermodynamic entropy S through:[23]

${\displaystyle S\ {\stackrel {\mathrm {def} }{=}}\ -VkH+{\text{constant}}}$

So, according to the H-theorem, S can only increase.

In quantum statistical mechanics (which is the quantum version of classical statistical mechanics), the H-function is the function:[24]

${\displaystyle H=\sum _{i}p_{i}\ln p_{i},\,}$

where summation runs over all possible distinct states of the system, and p_i is the probability that the system could be found in the i-th state.

This is closely related to the entropy formula of Gibbs,

${\displaystyle S=-k\sum _{i}p_{i}\ln p_{i}\;}$

and we shall (following e.g., Waldram (1985), p. 39) proceed using S rather than H.

First, differentiating with respect to time gives

${\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=-k\sum _{i}\left({\frac {dp_{i}}{dt}}\ln p_{i}+{\frac {dp_{i}}{dt}}\right)\\&=-k\sum _{i}{\frac {dp_{i}}{dt}}\ln p_{i}\\\end{aligned}}}$

(using the fact that ∑ dp_i/dt = 0, since ∑ p_i = 1, so the second term vanishes. We will see later that it will be useful to break this into two sums.)

Now Fermi's golden rule gives a master equation for the average rate of quantum jumps from state α to β; and from state β to α. (Of course, Fermi's golden rule itself makes certain approximations, and the introduction of this rule is what introduces irreversibility. It is essentially the quantum version of Boltzmann's Stosszahlansatz.) For an isolated system the jumps will make contributions

${\displaystyle {\begin{aligned}{\frac {dp_{\alpha }}{dt}}&=\sum _{\beta }\nu _{\alpha \beta }(p_{\beta }-p_{\alpha })\\{\frac {dp_{\beta }}{dt}}&=\sum _{\alpha }\nu _{\alpha \beta }(p_{\alpha }-p_{\beta })\\\end{aligned}}}$

where the reversibility of the dynamics ensures that the same transition constant ν_αβ appears in both expressions.

So

${\displaystyle {\frac {dS}{dt}}={\frac {1}{2}}k\sum _{\alpha ,\beta }\nu _{\alpha \beta }(\ln p_{\beta }-\ln p_{\alpha })(p_{\beta }-p_{\alpha }).}$

The two differences terms in the summation always have the same sign. For example:

${\displaystyle {\begin{aligned}w_{\beta }<w_{\alpha }\end{aligned}}}$

then

${\displaystyle {\begin{aligned}\ln w_{\beta }<\ln w_{\alpha }\end{aligned}}}$

so overall the two negative signs will cancel.

Therefore,

${\displaystyle \Delta S\geq 0\,}$

for an isolated system.

The same mathematics is sometimes used to show that relative entropy is a Lyapunov function of a Markov process in detailed balance, and other chemistry contexts.