Consider the energy and fluctuation in energy of an arbitrary system in contact with a heat rese

Consider the energy and fluctuation in energy of an...

Key Concepts

Partition Function

The partition function is a central quantity in statistical mechanics, defined as the sum over all possible states of the system weighted by the Boltzmann factor. It encapsulates all the statistical properties of a system in thermal equilibrium and is used to calculate macroscopic thermodynamic quantities. By taking logarithms and differentiating with respect to parameters like ?, relationships for quantities such as the average energy and entropy can be derived.

Thermodynamic Derivatives

Thermodynamic derivatives involve taking derivatives of the logarithm of the partition function with respect to thermodynamic variables like ? (inverse temperature). These derivatives yield important macroscopic quantities; for example, the first derivative with respect to ? gives the average energy, while higher derivatives provide information on energy fluctuations. This technique links microscopic statistics to observable thermodynamic properties.

Energy Fluctuations

Energy fluctuations refer to the variance in the energy of a system, defined as the difference between the average of the square of the energy and the square of the average energy. This concept is crucial for understanding the stability and response of a system in the canonical ensemble, and it is mathematically connected to the second derivative of the logarithm of the partition function with respect to ?.

Heat Capacity Relationship

Heat capacity is a measure of how much energy is required to change the temperature of a system by a given amount, and it is directly related to the system's energy fluctuations. In the canonical ensemble, the dispersion in energy is proportional to the temperature squared multiplied by the heat capacity, linking the microscopic energy fluctuations to the macroscopic thermal response of the system.

Ideal Gas Model

The ideal gas model, particularly the ideal monatomic gas, provides a simplified framework in which particles do not interact and only translational kinetic energy is considered. This model allows for explicit calculations of energy, energy fluctuations, and heat capacity, making it an important example for illustrating the connection between microscopic statistical properties and macroscopic observables such as the relative dispersion of energy.

Transcript

00:01 Okay, so in this video, i'm going to be talking about problem number 18 of chapter 6.

00:08 There's two results that we want to prove.

00:10 So let's go ahead and prove the first one.

00:13 So the first one is going to be to prove that the average of energy squared is equal to 1 over z of d2z, d, beta squared, where beta is 1 over k t okay so let's prove this the average of of a variable is equal there's a very standard formula for computing averages of variables and that formula is this it is a it's a sum over states energy of that state squared times the probability of that state.

01:01 That's the formula for average.

01:05 Now, now the probability of a state can be written as the boltzman factor over the partition function.

01:13 So let's write it, let's write it as a sum over states of energy of that state squared multiplied by the boltzman factor, which is a exponential of negative energy of that state over, over tau.

01:32 Let me write the 1 over tau as as beta, because that's the letter that we're using.

01:39 Over the partition function, which is a function of temperature.

01:44 Okay.

01:45 So i'm going to pull the 1 over z out of the summation because it does not, the 1 over z is a function of temperature.

01:54 It does not depend on which state we're talking about.

01:56 So i'm going to pull it out of the, out of the summation.

01:59 So i have 1 over z multiplied by a summation of your states, energy of that state squared, e to the negative beta energy of that state.

02:17 Now, as you can clearly see, every time you differentiate this exponential factor with respect to beta, you get a factor of negative es in the front.

02:31 So if you differentiate it twice, you get two factors of negative es, right? you get two of them.

02:40 So this just becomes es squared.

02:43 In other words, this factor here can be obtained by differentiating twice.

02:55 Okay, you can clearly verify that if you differentiate, differentiate this exponential twice with respect to beta you get es squared times the exponential.

03:09 Alright? now the property of a derivative is that i take the summation of a derivative is the that's the same thing as the derivative of a summation.

03:20 That's why i can pull this derivative outside of the summation.

03:25 So now i have 1 over z times second derivative of a sum over states of both main factors.

03:39 Now this summation, that's just the partition function.

03:45 So i'm going to write it as the partition function.

03:52 Okay? now that completes the proof of the first result.

04:00 Let's move on to the second result.

04:03 We want to prove that the standard deviations of energy of the system is equal to k t square root heat capacity over over k where k is just the boltman constant and c is the heat capacity which is in this case defined as the partial derivative of average energy with respect to temperature okay so let's go ahead and prove that let's first of all take the partial derivative of the average energy with respect to temperature.

04:52 Okay.

04:54 Now what is this? this is actually, this is the partial derivative with respect to temperature of the average energy, but the average energy, there's an expression for it, so i'm going to use that expression.

05:09 Negative d by d beta of natural log of the partition function we have this expression from from a previous problem and and and and we're allowed to use this expression because the problem clearly says that we can use the results from previous from we can use previous results anyway we have we have negative d by d t d by d beta log partition function.

05:44 Now, what is d by d t? let me remind you of the relation that connects beta to t.

05:52 Beta is defined as 1 over kt.

05:56 Now, i want to take this partial derivative with respect to t.

06:04 I'm going to change it to something that i'm going to change it to a derivative with respect to beta.

06:10 How do i do that? well, the partial derivative with respect to t, that's the same thing as partial derivative with respect to beta times d beta d .t.

06:22 That's just the chain rule, multiplied by d by the beta, log z.

06:30 Okay.

06:31 Now, this is just a number.

06:34 D, b, dt is just a number.

06:36 So that's why i can change the order here.

06:41 Multiplied by d squared, d b squared, excuse me d beta squared, log of the partition function.

06:52 Now, what is d beta d t? let's go ahead and calculate d beta d t.

06:58 Well, d beta d t, according to this expression here, it's just 1 over k, negative 1 over t squared.

07:07 That's what the beta d t is.

07:09 Okay, so i'm going to take this, plug it into here...