- #1
Sam_Goldberg
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Hi guys, I am reading Reif's book on statistical mechanics and have a question on the H theorem. In section 2.3, Reif gives (on page 54) both the definition of equilibrium as well as a fundamental postulate.
Definition: "An equilibrium situation is characterized by the fact that the probability of finding the system in anyone state is independent of tiem (i.e., the representative ensemble is the same irrespective of time). All macroscopic parameters describing the isolated system are then also time-independent."
Fundamental Postulate: "An isolated system in equilibrium is equally likely to be in any of its accessible states."
A few pages later Reif talks about how nonequilibrium situations tend to approach equilibrium, and then refers me to his appendix A.12 on the H theorem (pages 624-626). His arguments are essentially those on this wikipedia page (read up to deltaS>=0):
http://en.wikipedia.org/wiki/H-theorem#Quantum_mechanical_H-theorem
Here is the question: look at the forumula for dS/dt before deltaS>=0. It appears as if dS/dt is equal to zero if and only if (correct?) Pa = Pb for all a and all b, since each term in the double summation contributes a positive value for dS/dt unless the probabilities are equal. It therefore seems as if one could prove the fundamental postulate I listed above as follows: if the system is isolated and in equilibrium, then the probabilities Pi do not change in time by virture of Reif's definition of equilibrium. Thus, in particular, H (or S = -kH) does not change in time. However, dH/dt (or dS/dt) is zero if and only if all the probabilities are equal, and, therefore, we may conclude that isolation and equilibrium implies that the system is equally likely to be in its possible states. This "proof" cannot be correct, for Reif emphasized that the fundamental postulate cannot be proved; it is an axiom on which statistical mechanics is based. Thus, how is the above incorrect? Where did I go wrong?
Thanks in advance for all your help. (By the way, how do you pronounce the author's name?)
Definition: "An equilibrium situation is characterized by the fact that the probability of finding the system in anyone state is independent of tiem (i.e., the representative ensemble is the same irrespective of time). All macroscopic parameters describing the isolated system are then also time-independent."
Fundamental Postulate: "An isolated system in equilibrium is equally likely to be in any of its accessible states."
A few pages later Reif talks about how nonequilibrium situations tend to approach equilibrium, and then refers me to his appendix A.12 on the H theorem (pages 624-626). His arguments are essentially those on this wikipedia page (read up to deltaS>=0):
http://en.wikipedia.org/wiki/H-theorem#Quantum_mechanical_H-theorem
Here is the question: look at the forumula for dS/dt before deltaS>=0. It appears as if dS/dt is equal to zero if and only if (correct?) Pa = Pb for all a and all b, since each term in the double summation contributes a positive value for dS/dt unless the probabilities are equal. It therefore seems as if one could prove the fundamental postulate I listed above as follows: if the system is isolated and in equilibrium, then the probabilities Pi do not change in time by virture of Reif's definition of equilibrium. Thus, in particular, H (or S = -kH) does not change in time. However, dH/dt (or dS/dt) is zero if and only if all the probabilities are equal, and, therefore, we may conclude that isolation and equilibrium implies that the system is equally likely to be in its possible states. This "proof" cannot be correct, for Reif emphasized that the fundamental postulate cannot be proved; it is an axiom on which statistical mechanics is based. Thus, how is the above incorrect? Where did I go wrong?
Thanks in advance for all your help. (By the way, how do you pronounce the author's name?)