I would like to share my thoughts and questions on the issue. The Boltzmann H theorem based on classical mechanics is well discussed in various literatures, the irreversibility comes from his assumption of molecular chaos, which cannot be justified from the underlying dynamical equation. Here I will try to say something on quantum H theorem, the point I want to make is that, although seemingly H theorem can be derived from unitarity, the true entropy increase in fact comes from the non-unitary part of quantum mechanics. Let me first recap the derivation using unitarity $^{1,2}$.
H theorem as a consequence of unitarity
Denote by $P_k$ the probability of a particle appearing on the state $|k\rangle$, $A_{kl}$ the transition rate from state $|k\rangle$ to state $|l\rangle$, then by the master equation
$${\frac {dP_{k}}{dt}}=\sum _{l}(A_{{kl }}P_{l }-A_{{l k}}P_{k})=\sum _{{l\neq k}}(A_{{kl }}P_{l }-A_{{l k}}P_{k})\cdots\cdots(1).$$
Then we take the derivative of entropy
$$S=-\sum_k P_k\ln P_k\cdots\cdots(2),$$
we obtain
$$\frac{dS}{dt}=-\sum_k\frac{dP_k}{dt}\left(1+\ln P_k\right)\cdots\cdots(3).$$
Together with (1) we have
$$\frac{dS}{dt}=-\sum_{kl}\left\{(1+\ln P_k)A_{{kl }}P_{l }-(1+\ln P_k)A_{{l k}}P_{k}\right\}\cdots(4).$$
For the seond second term let us interchange the dummy indices $k$ and $l$, we get
$$\frac{dS}{dt}=\sum_{kl}(\ln P_l-\ln P_k)A_{kl}P_l\cdots\cdots(5)$$
Now use the mathematical identity $(\ln P_l-\ln P_k)P_l\geq P_l- P_k$, we obtain
$$\frac{dS}{dt}\geq \sum_{kl}(P_l-P_k)A_{kl}= \sum_{kl}P_l(A_{kl}-A_{lk})\\=\sum_{l}P_l\big\{\sum_{k}(A_{kl}-A_{lk})\big\}\cdots\cdots(6)$$
Now unitarity ensures $\sum_{k}A_{kl}$ and $\sum_{k}A_{lk}$ are both 0, because as transition rates,
$$\sum_{k}A_{kl}=\frac{d}{dt}\sum_{k}|\langle l|S|k\rangle|^2=\frac{d}{dt}\sum_{k}\langle l|S|k\rangle\langle k|S^{\dagger}|l\rangle\\=\frac{d}{dt}\langle l|SS^{\dagger}|l\rangle=\frac{d}{dt}\langle l|l\rangle=0\cdots\cdots(7),$$
where $S$ is the unitary time evolution operator describing the system. This is nothing but saying the total transition probability from one state to all states must be 1. It is clear (6) and (7) imply the H theorem:
$$\frac{dS}{dt}\geq 0.$$
Where does the irreversibility come from?
Now we are in a position to question ourselves with Loschmidt's paradox, analogously to its classical version: There are many unitary and time-reversible quantum mechanical systems, if we have just derived H theorem using unitarity alone, how can it be reconciled with time-reversibility of the underlying dynamics?
What sneaked into the above derivation?
The crucial thing to notice is that, in the quantum regime, the definition of entropy using equation (2) is inherently an impossible one: the value of the entropy in (2) depends on the basis we choose to describe the system!
Consider a two-level system with two choices of orthogonal basis $\{|1\rangle, |2\rangle\}$ and $\{|a\rangle, |b\rangle\}$ related by
$$|1\rangle=\frac{1}{\sqrt2}(|a\rangle+|b\rangle),\\|2\rangle=\frac{1}{\sqrt2}(|a\rangle-|b\rangle).$$
Suppose the system is in the state $|1\rangle$, then the entropy formula gives $S=0$ in the first choice of basis since it has 100% chance to appear in $|1\rangle$, while in the other basis $S=\ln2$ since it has 50%-50% chance to appear in either $|a\rangle$ or $|b\rangle$.
Now we may argue, it is one thing that to say the system is in $\frac{1}{\sqrt2}(|a\rangle+|b\rangle)$ and have the potential 50%-50% chance to transit into $|a\rangle$ and $|b\rangle$ after a measurement, but a different thing to say the transition has been realized by some measurement. Two situations must be described differently. If we look back to our derivation, it is not hard to see what we really did was, after a basis state evolves to a new state which is a superposition of the basis states, we assumed transitions to original basis states have happened instead of just staying in that superposition state, and in fact the original definition of entropy is not capable of describing such situation, as explained just now.
A plausible definition of quantum entropy is the Von Neumann entropy, which is a basis-independent definition of entropy, and in this description, the entropy of a unitarily evolving system is constant in time, while a (projective) measurement can increase the entropy.
Based on the above comparison, we see the irreversibility really comes as an assumption, the assumption that a measurement/decoherence has happened, and as we know, a (projective) measurement is a non-unitary, irreversible process, no paradox anymore.
My own question on the issue is, what to make of the fact that von Neumann entropy is constant in time? Does it mean it is incapable of describing a closed system evolving from non-equilibrium to equilibrium, or should we just reverse the argument and say any non-equilibrium to equilibrium evolution must be described by some non-unitary process?
1.Rephrased from section 3.6 of The Quantum Theory of Fields, Vol1, S. Weinberg
2.If I remember correctly(which I'm not quite confident on), such derivation was first given by Pauli, and he correctly spotted the origin of irreversibility, which he called the "random phase assumption".
This post imported from StackExchange Physics at 2016-08-04 12:34 (UTC), posted by SE-user Jia Yiyang
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