Multiplicity of a Large Einstein Solid

1. Multiplicity of a Large Einstein Solid We consider the high-temperature case of q≫N. How sharp is the multiplicity function?

2. Concepts of Statistical Mechanics The upshot is that as N and q become large, the multiplicity function peak becomes narrower and narrower. The most probable macrostate becomes more and more probable relative to the other possible macrostates. Put another way, fluctuations from the most probable macrostate are very small in large systems. • The macrostate is specified by a sufficient number of macroscopically measurable parameters (for an Einstein solid – N and U). • The microstate is specified by the quantum state of each particle in a system (for an Einstein solid – # of the quanta of energy for each of N oscillators) • The multiplicity is the number of microstates in a macrostate. For each macrostate, there is an extremely large number of possible microstates that are macroscopically indistinguishable. • The Fundamental Assumption: for an isolated system, all accessible microstate are equally likely. • The probability of a macrostate is proportional to its multiplicity. This will sufficient to explain irreversibility.

3. The ideal gas So far we have treated quantum systems whose states in the configuration (phase) space may be enumerated. When dealing with classical systems with translational degrees of freedom, we need to learn how to calculate the multiplicity. Multiplicity for a single particle in a box particle in a one-dimensional “box” -L L The total number of ways of filling up the cells in phase space is the product of the number of ways the “space” cells can be filled times the number of ways the “momentum” cells can be filled. Quantum mechanics (Heisenberg uncertainty principle; Appendix A) helps us to numerate all different states in the configuration (phase) space: px Lp -L L x px -Lp x Q.M. The number of microstates:

4. n p For a molecule in a three-dimensional box: the state of the molecule is a point in the 6D space - its position (x,y,z) and its momentum (px,py,pz). The number of “space” microstates is: For N molecules: There is some momentum distribution of molecules in an ideal gas (Maxwell), with a long “tail” that goes all the way up to p = (2mU)1/2 (U is the total energy of the gas). However, the momentum vector of an “average” molecule is confined within a sphere of radius p ~ (2mU/N)1/2 (U/N is the average energy per molecule). Thus, for a single “average” molecule: The total number of microstates for N molecules: However, we have overcounted the multiplicity, because we have assumed that the atoms are distinguishable. For indistinguishable quantum particles, the result should be divided by N! (the number of ways of arranging N identical atoms in a given set of “boxes”):

5. pz py px How to compute Vp? Momentum constraints: 1 particle - 2 particles - The accessible momentum volume for N particles = the “area” of a 3N-dimensional hyper-sphere in the momentum space N =1 Monatomic ideal gas: (3N degrees of freedom) f N- the total # of “quadratic” degrees of freedom

6. Entropy Entropy is a measure of how evenly energy is distributed in a system. In a physical system, entropy provides a measure of the amount of energy that cannot be used to do work. The entropy of a particular macrostate is defined as: The Second Law of Thermodynamics says: Systems tend to evolve in the direction of increasing multiplicity. That is, entropy tends to increase. The most general interpretation of entropy is as a measure of our uncertainty about a system. The equilibrium state of a system maximizes the entropy because we have lost all information about the initial conditions except for the conserved variables; maximizing the entropy maximizes our ignorance about the details of the system.This uncertainty is not of the everyday subjective kind, but rather the uncertainty inherent to the experimental method and interpretative model. The second law of thermodynamics can be stated to say that the entropy of an isolated system always increases, and such processes which increase entropy can occur spontaneously. Since entropy increases as uniformity increases, the second law says qualitatively that uniformity increases. The total entropy of two interacting systems, such as the two Einstein solids, is

7. Maxwell's demon Maxwell's demon is a thought experiment created by James Clerk Maxwell in 1867 to "show that the Second Law of Thermodynamics has only a statistical certainty". It demonstrates Maxwell's point by hypothetically describing how to violate the Second Law: a container is divided into two parts by an insulated wall, with a door that can be opened and closed by what came to be called "Maxwell's demon". The demon opens the door to allow only the "hot" molecules of gas to flow through to a favoured side of the chamber, causing that side to gradually heat up while the other side cools down, thus decreasing entropy. Note that the demon must allow molecules to pass in both directions in order to produce only a temperature difference; one-way passage only of faster-than-average molecules from A to B will cause higher temperature and pressure to develop on the B side; the demon must actually let more slow molecules pass from B to A than fast ones pass from A to B in order to make B hotter at the same pressure. Actually, the second law of thermodynamics will not actually be violated, if a more complete analysis is made of the whole system including the demon. To explain the paradox scientists point out that to realize such a possibility the demon would still need to use energy to observe the molecules (in the form of photons for example). And the demon itself (plus the trap door mechanism) would gain entropy from the gas as it moved the trap door. Thus the total entropy of the system still increases.