(a) Give the definition of entropy in statistical physics....

(a) Give the definition of entropy in statistical physics. (b) Give a general argument to explain why and under what circumstances the entropy of an isolated system $A$ will remain constant, or increase. For convenience you may assume that $A$ can be divided into subsystems $B$ and $C$ which are in weak contact with each other, but which themselves remain in internal thermodynamic equilibrium.

(a) Give the definition of entropy in statistical physics.
(b) Give a general argument to explain why and under what circumstances the entropy of an isolated system $A$ will remain constant, or increase. For convenience you may assume that $A$ can be divided into subsystems $B$ and $C$ which are in weak contact with each other, but which themselves remain in internal thermodynamic equilibrium.

Key Concepts

Statistical Entropy

This concept defines entropy as a measure of the number of microscopic configurations (microstates) corresponding to a macroscopic state. In statistical physics, it is typically expressed via Boltzmann’s formula, S = k_B ln(?), where k_B is Boltzmann's constant and ? is the number of accessible microstates. This definition bridges the gap between microscopic behavior and macroscopic thermodynamic properties.

Second Law of Thermodynamics

This principle states that within an isolated system, the total entropy can never decrease over time. It explains the natural progression towards disorder or equilibrium in a system. The law underlines that irreversible processes, such as spontaneous heat flow, result in an overall increase in entropy, embodying the fundamental directional flow of time in thermodynamic processes.

Isolated Systems

An isolated system is one that does not exchange matter or energy with its surroundings. For such systems, any changes in entropy arise only from internal processes. The properties of isolated systems are central in thermodynamics because their behavior is governed strictly by intrinsic dynamics, making them ideal scenarios for applying the second law without external influence.

Thermodynamic Equilibrium and Weak Coupling

This concept involves systems (or subsystems) that are in a state where macroscopic properties remain constant over time, and no net macroscopic flows of matter or energy occur. The idea of weak coupling or weak interactions between subsystems allows them to reach internal equilibrium independently while still being part of a larger system. Perturbations or energy exchanges that momentarily disturb this balance eventually lead the system back to a state of equilibrium, often accompanied by an overall increase in entropy if the process is irreversible.

Transcript

00:01 All right, so part a.

00:02 So we're assuming that the total number of microstates accessible to a system is omega.

00:08 And we're going to show that the entropy of that system is the maximum when all microstates are equally likely to occur.

00:16 Okay, so entropy s is given to us by the boltzman entropy formula.

00:23 Negative kb, sigma, which is just adding everything up all over r, p .r.

00:32 Ln of pr, where pr is the probability of the system being in that microstate r.

00:41 So that's the prob of being in a particular microstate, which is just like what the system is doing.

00:54 In kb, that's just the boltzman constant.

00:57 So if there are omega microstates and they are all equally probable, then that means that pr, equals 1 over omega for every single one of those individual microstates.

01:15 So if we plug this guy into our formula, we see the s equals negative kb, sigma r, 1 over omega, ln of 1 over omega.

01:29 But since there are omega terms in this sum, this means that we can rewrite it as kb, omega, comes 1 over omega, ln, of 1 over omega, ln, of.

01:41 Of 1 over omega.

01:44 We rearrange and we get negative kb, ln of 1 over omega, and properties of log says that this is positive kb, ln of omega, and this means that your entropy is maximized.

02:05 So s is maxed, when all of those microstates are equally probable.

02:16 This also aligns with something called the max, or maximum entropy principle in statistical physics and statistical mechanics.

02:28 Okay, okay.

02:29 Part b.

02:31 If on the other hand, we have some ensemble.

02:35 And in this case, we're sharing some energy with a mean energy of e.

02:46 Okay.

02:48 Mean value e bar.

02:51 Show that the entropy is given by the same formal expression, and maximize when pr goes as a...

02:56 Exponential negative beta er.

02:59 Ok -dokey.

03:01 So let's go ahead and start out with that entropy expression.

03:06 So here we go.

03:07 S equals negative kb, sigma sum of pr, ln of pr.

03:15 And in this case, another way of saying that this mean energy thing right here that they all share, we say that your energy is constrained.

03:26 So your energy in this case is constrained.

03:30 The ensemble, has that fixed energy, mean energy.

03:35 So in order to maximize your entropy subject to this constraint, so we're trying to max with subject to this constraint of that fixed energy, we do the following.

03:50 Start out with that entropy expression.

03:52 Now our constraints, well, of course, we must normalize everything.

03:59 So if you add all those probabilities out for each of those microstates, they better all equal to one.

04:04 And your energy, is going to be that the sum of each of those energies for each microstate better all equal when you add them together to that average energy.

04:18 So we're going to use the lagrange multipliers.

04:21 So that method says that we define some other letter, some other thing...

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