# Deriving Statistical Mechanics laws from Quantum Mechanics?

Since the law of individual molecule is governed by Quantum Mechanics, and the interaction of large number of molecule is governed by Statistical Mechanics, can we derive Statistical Mechanics from Quantum Mechanics?

For a rigorous derivation of statistical mechanics from quantum mechanics see Chapters 8-10 of my book: Classical and Quantum Mechanics via Lie algebras http://lanl.arxiv.org/abs/0810.1019

• How do you get around conservation of entropy? – Ron Maimon Jun 2 '12 at 9:22
• Chapter 10 discusses description levels, which accounts for the approximations needed to make the thermal description in terms of fewer than all observables work. The set of observables deemed sufficient for the description determines the accuracy with which the dynamics is reproduced. (For an exact dynamics one needs all microscopic observables, which means one leaves the realm of statistical mechanics.) – Arnold Neumaier Jun 2 '12 at 9:28
• I really don't understand how people publish books as pdf without a working interactive index. Also this is not an answer. – pindakaas Apr 24 '15 at 7:01

We certainly apply statistical mechanics to quantum systems. In fact, that is where it most often comes up in a useful way. Consider some canonical examples like the Ising model, blackbody radiation, and Bose-Einstein condensation.

But the power of statistical mechanics is that it is formulated in a manner that is independent of the kinematics of the systems being described. All it relies on an enumeration of the microstates a system can be in, and the energy associated with each microstate. It does not deal with the details of the equations of motion governing the system (i.e., the Schrodinger equation is not used explicitly). Statistical mechanics provides a framework for tracking the flow of information (entropy) in a system using laws of probability, and in that sense it is too fundamental to be derived from a theory like quantum mechanics.

EDIT: I should clarify that the principles of statistical mechanics on their own are not completely sufficient to describe a system of many particles. Some details of quantum mechanics will be needed in order to enumerate the microstates and their energy. The point is that even if the laws of quantum mechanics were slightly different, the principles of statistical mechanics would still be used in the same way to describe the many-particle system.

• I think we will find, interestingly, that it is not completely independent of the kinematics. For example, one kinematics leads to a trivial stochastic process, and in order to get Brownian motion, you need a radically different kinematics. I have conjectured, and am working on it, that the difference between positive temperature and negative temperature will also be something like that. Macroscopic properties are robust to «small» changes in the kinematics, e.g., that between hard spheres and Coulomb forces, but radical changes produce different «phases», in a new sense of the word. – joseph f. johnson Feb 12 '13 at 0:42

The basics of statistical mechanics are no more than ordinary probability theory so it's pure maths which can be applied to any abstract system like a set or marbles. In other words, statmech is a handy tool to apply probability theory to a system, but it doesn't contain any new physical law. Therefore you obviously cannot derive it from QM, but you can check if QM satisfies the presuppositions of statistical mechanics and hence apply statmech to QM.

• Well, the fundamental postulate is some input. – Nikolaj-K Jun 2 '12 at 19:59
• True, that's what I meant by "presuppositions". You can apply the methods of statmech to anything where (micro)states have equal probability. Entropy laws follow from that. You could equally apply statmech to a set of cards or anything abstract. – Gerenuk Jun 2 '12 at 20:06
• Well, my own hobbyhorse is the exact contrary to your assertions. I would like to make a kind of quantum statistical mechanics, not the usual one which has a quantum limit but a new one with a classical limit, be the basis for probability theory, and not the other way round. See my published worldscientific.com/doi/pdf/10.1142/9789812702340_0017 or at arxiv.org/abs/quant-ph/0508059 At about the same time, or a year earlier, Prof. von Plato of Helsinki Univ. made some progress in the same direction for classical physics. – joseph f. johnson Feb 12 '13 at 0:53

Already in the 1920s Darwin and Fowler started with the Quantum Theory as their basis, and went on to derive the usual statistical mechanical results rigorously. Now in principle, you could do the analogous thing even without the assumption that each microstate had equal probability: you could postulate any rule for «weighting» the micro-states you want, try to imitate their calculations, and compare the results with experiment, which is what they did for their particular rule, the usual one mentioned here, of weighting each state equally (they explained why they would not use the word «probability» but instead the neutral term, «weight».

In the 1940s, Schroedinger complained about the assumption that a microstate was the occupation of an energy level, i.e., that the microstates were (basically) the stationary states. He said that to be really logical, one would have to treat any, even time-varying, pure state given by a wave function as a possible micro-state, and then show that you would get the same answers as you did by making the traditional assumption of a discrete (basis) set of stationary micro-states. But he, plaintively, admitted that that task was still impossible, so he was going to teach the usual method in spite of this defect.

I strongly recommend learning the foundations of Statistical Mechanics from Schroedinger's little volume of Dublin lectures, except the last chapter, (for the physicist's point of view) plus the first half of Khintchine's book (amazingly, published in Moscow in 1943) The Mathematical Foundations of Statistical Mechanics (but skipping almost all the proofs, of course).