How Can Internal Energy of the Canonical Ensemble Change (Fluctuate)?

[Dario56]

Canonical ensemble is the statistical ensemble which is applicable for the closed system in contact with the reservoir at constant temperature $T$. Canonical ensemble is characterized by the three fixed variables; number of particles $N$, volume $V$ and temperature $T$.

What is said is that the internal energy of the system described by the canonical ensemble can change as heat can be exchanged with the reservoir. Therefore, we can calculate expectation value of internal energy by taking into account probability and energy of all possible microstates of the system.

However, if the system and reservoir are at thermal equilibrium, how can any heat be exchanged? For heat to flow, temperature difference should exist.

Also, if we for example had an one component and one phase system (like an ideal gas or a pure liquid), we know that its state is completely specified by two intensive variables (in context of the canonical ensemble; molar volume and temperature) and number of particles (number of moles). Since all these variables are specified in the canonical ensemble, internal energy and all other thermodynamic variables of the system are also specified.

If this is the case, how can internal energy change?

 

[vanhees71]

In the canonical ensemble the system is in contact with a heat reservoir and thus always exchanges energy. What's fixed in the canonical ensemble is not the energy of the system but the mean energy. Due to the exchange of energy with the reservoir there are fluctuations of the energy of the system. It's calculated as follows:

The canonical statistical operator is
$$\hat{\rho} = \frac{1}{Z} \exp(-\beta \hat{H}).$$
The partition some is
$$Z=\mathrm{Tr} \exp(-\beta \hat{H}),$$
and $\beta=1/(k T)$ (with $k$ the Boltzmann constant).
The average energy is
$$U=\langle H \rangle=-\partial_{\beta} \ln Z=\frac{\mathrm{Tr}(\hat{H} \hat{\rho})}{Z}.$$
Take another derivative wrt. $\beta$, and you get
$$\partial_{\beta} U = -\frac{\mathrm{Tr} \hat{H}^2 \hat{\rho}}{Z} - \frac{\partial_{\beta} Z}{Z^2} \mathrm{Tr}(\hat{H} \hat{\rho}) = -\langle H^2 \rangle + \langle H \rangle^2=-\Delta U^2.$$
Now
$$\partial_{\beta} U = \frac{\mathrm{d} T}{\mathrm {d} \beta} \frac{\partial U}{\partial T} = -\frac{1}{k T^2} \partial_T U = -\frac{1}{k T^2} C_V,$$
i.e.,
$$C_V = \frac{1}{k_B T^2} \Delta U^2.$$
The specific heat is thus proportional to the variance of the energy.

 

[Dario56]

In the canonical ensemble the system is in contact with a heat reservoir and thus always exchanges energy. What's fixed in the canonical ensemble is not the energy of the system but the mean energy. Due to the exchange of energy with the reservoir there are fluctuations of the energy of the system. It's calculated as follows:

The canonical statistical operator is
$$\hat{\rho} = \frac{1}{Z} \exp(-\beta \hat{H}).$$
The partition some is
$$Z=\mathrm{Tr} \exp(-\beta \hat{H}),$$
and $\beta=1/(k T)$ (with $k$ the Boltzmann constant).
The average energy is
$$U=\langle H \rangle=-\partial_{\beta} \ln Z=\frac{\mathrm{Tr}(\hat{H} \hat{\rho})}{Z}.$$
Take another derivative wrt. $\beta$, and you get
$$\partial_{\beta} U = -\frac{\mathrm{Tr} \hat{H}^2 \hat{\rho}}{Z} - \frac{\partial_{\beta} Z}{Z^2} \mathrm{Tr}(\hat{H} \hat{\rho}) = -\langle H^2 \rangle + \langle H \rangle^2=-\Delta U^2.$$
Now
$$\partial_{\beta} U = \frac{\mathrm{d} T}{\mathrm {d} \beta} \frac{\partial U}{\partial T} = -\frac{1}{k T^2} \partial_T U = -\frac{1}{k T^2} C_V,$$
i.e.,
$$C_V = \frac{1}{k_B T^2} \Delta U^2.$$
The specific heat is thus proportional to the variance of the energy.

Yes, I think I've got it in the meantime. Heat can be exchanged because (and so internal energy changes) thermal contact exists between the system and the reservoir which means that particles in the system and reservoir can collide and thus exchange energy. It is the number of microstates for certain energy level of the system which determines how likely is the system to be found with that energy.

Very important thing to note is that as number of particles approach infinity, variance of energy tends to zero and thus statistical thermodynamics becomes unnecessary to describe thermodynamic state of the real systems as number of particles is usually huge.

In another words, energy of the system is practically certain to be found at an expectation value of all possible microstates.

 

[Lord Jestocost]

There seems to be some misunderstanding about the “canonical ensemble” approach. The idea behind is well explained in “THERMODYNAMICS AND STATISTICAL MECHANICS” by Walter Greiner, Ludwig Neise and Horst Stöcker, chapter 7 “The Canonical Ensemble”:

“Therefore we now want to reflect on the probability distribution (phase-space density) of a system at a given temperature (a system $S$ in a heat bath $R$). To do this, we apply what we have learned up to now about the total closed system (heat bath plus system). The total energy of the whole system

$E=E_R+E_S ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(7.1)$

has a constant given value. By definition, the heat bath is very large compared to the system itself, so that

$E_S/E=(1-E_R/E) \ll 1 ~~~~~~~~~~(7.2)$

Since now it is no longer the energy $E_S$ which is fixed, but the temperature, the system $S$ will be able to assume all possible microstates $i$ with different energies $E_i$ with a certain probability distribution. However, we expect that microstates with very large $E_i$ will appear only very scarcely. We ask for the probability $p_i$ of finding the system $S$ in a certain microstate $i$ with the energy $E_i$.”

 

[Dario56]

In the canonical ensemble the system is in contact with a heat reservoir and thus always exchanges energy. What's fixed in the canonical ensemble is not the energy of the system but the mean energy. Due to the exchange of energy with the reservoir there are fluctuations of the energy of the system. It's calculated as follows:

The canonical statistical operator is
$$\hat{\rho} = \frac{1}{Z} \exp(-\beta \hat{H}).$$
The partition some is
$$Z=\mathrm{Tr} \exp(-\beta \hat{H}),$$
and $\beta=1/(k T)$ (with $k$ the Boltzmann constant).
The average energy is
$$U=\langle H \rangle=-\partial_{\beta} \ln Z=\frac{\mathrm{Tr}(\hat{H} \hat{\rho})}{Z}.$$
Take another derivative wrt. $\beta$, and you get
$$\partial_{\beta} U = -\frac{\mathrm{Tr} \hat{H}^2 \hat{\rho}}{Z} - \frac{\partial_{\beta} Z}{Z^2} \mathrm{Tr}(\hat{H} \hat{\rho}) = -\langle H^2 \rangle + \langle H \rangle^2=-\Delta U^2.$$
Now
$$\partial_{\beta} U = \frac{\mathrm{d} T}{\mathrm {d} \beta} \frac{\partial U}{\partial T} = -\frac{1}{k T^2} \partial_T U = -\frac{1}{k T^2} C_V,$$
i.e.,
$$C_V = \frac{1}{k_B T^2} \Delta U^2.$$
The specific heat is thus proportional to the variance of the energy.

One thing I am not sure about is that the canonical ensemble is applicable for systems with constant temperature. Also, volume and number of particles are constant.

If system can exchange heat with the reservoir (internal energy of the system can change), how can temperature remain the same in every microstate given the fact that both volume and number of particles remain the same?

For the sake of simplicity, we can take ideal gas as our system. We know its internal energy depends only on temperature. If so, than exchanging heat with the reservoir (which happens in the canonical ensemble) should change its temperature.

This is problematic as in the canonical ensemble temperature of the system should be independent of the system's energy.

 

[vanhees71]

The canonical equilibrium ensemble by definition describes a system in contact with a large reservoir keeping it at constant temperature, i.e. the reservoir is so large that the exchange of energy doesn't change the temperature of the reservoir considerably. Volume and particle number are by definition fixed. In the maximum entropy principle they are external parameters, while the temperature occurs as a Lagrange parameter to fix the expectation value of the total energy.

 

[Dario56]

The canonical equilibrium ensemble by definition describes a system in contact with a large reservoir keeping it at constant temperature, i.e. the reservoir is so large that the exchange of energy doesn't change the temperature of the reservoir considerably. Volume and particle number are by definition fixed. In the maximum entropy principle they are external parameters, while the temperature occurs as a Lagrange parameter to fix the expectation value of the total energy.

Yes, however if heat is exchanged how can system stay at the same temperature if both volume and number of particles are constant?

In the simple case of ideal gas, this shouldn't really hold as only variable which defines temperature is the internal energy and it changes due to heat exchange with the reservoir.

 

[vanhees71]

It's approximately posssible if the reservoir is very much larger than the system coupled to it.

 

[Dario56]

It's approximately posssible if the reservoir is very much larger than the system coupled to it.

Hmm, well the reservoir should keep the temperature of the system constant as its heat capacity is much bigger than the system. However, in the canonical ensemble, heat can flow freely from and out of the system which changes its internal energy and thus temperature should change.

What does heat capacity of the reservoir have to do with that?

 

[Lord Jestocost]

One couples two systems in such a way that they can exchange energy. Then one makes one of the systems enormous, and calls it the reservoir, $R$ (also known as a heat bath). The reservoir is so large that one can take quite a lot of energy out of it and yet it can remain at essentially the same temperature, $T_R$. The temperature of the system of interest, $S$, isn’t constant; in dependence of $S$'s size, it fluctuates around an average of $T_R$, so to speak.

 

[vanhees71]

Hmm, well the reservoir should keep the temperature of the system constant as its heat capacity is much bigger than the system. However, in the canonical ensemble, heat can flow freely from and out of the system which changes its internal energy and thus temperature should change.

What does heat capacity of the reservoir have to do with that?

The system is very small compared to the heat bath, i.e., the little energy it can exchange with the heat bath is negligible compared to the (mean) energy of the heat bath.

 

[Lord Jestocost]

On basis of a notification by @bob012345, one should specify what is meant by expressions like ‘the temperature is fixed” or ‘at a given temperature’ in context with the canonical ensemble approach. If a system $S_1$ is thermally coupled to a system $S_2$, one has for the number of microstates which can realize a configuration with $E_1$ and $E_2=E_t-E_1$
$$\Omega(E_t)=\Omega_1(E_1) \Omega_2(E_2)=\Omega_1(E_1) \Omega_2(E_t-E_1),$$
where $E_t=E_1+E_2$ is the total energy of the compound system.

The most likely division of energy between $S_1$ and $S_2$ is the one that maximizes $\Omega_1(E_1) \Omega_2(E_2)$ because this will correspond to the greatest number of possible microstates. For the most likely division of energy between the systems in thermal contact one has the condition
$$\frac{\partial \Omega_1(E_1)}{\partial E_1} = \frac{\partial \Omega_2(E_2)}{\partial E_2}$$
since it maximizes the total number of microstates. This division of energy is usually called ‘being at the same temperature’ and so one can identify $\frac{\partial \Omega(E)}{\partial E}$ with the temperature $T$ (so that $T_1 = T_2$).

The temperature $T$ can now be defined by
$$\frac{1}{k_BT}=\frac{\partial \Omega(E)}{\partial E}.$$

 

[Dario56]

On basis of a notification by @bob012345, one should specify what is meant by expressions like ‘the temperature is fixed” or ‘at a given temperature’ in context with the canonical ensemble approach. If a system $S_1$ is thermally coupled to a system $S_2$, one has for the number of microstates which can realize a configuration with $E_1$ and $E_2=E_t-E_1$
$$\Omega(E_t)=\Omega_1(E_1) \Omega_2(E_2)=\Omega_1(E_1) \Omega_2(E_t-E_1),$$
where $E_t=E_1+E_2$ is the total energy of the compound system.

The most likely division of energy between $S_1$ and $S_2$ is the one that maximizes $\Omega_1(E_1) \Omega_2(E_2)$ because this will correspond to the greatest number of possible microstates. For the most likely division of energy between the systems in thermal contact one has the condition
$$\frac{\partial \Omega_1(E_1)}{\partial E_1} = \frac{\partial \Omega_2(E_2)}{\partial E_2}$$
since it maximizes the total number of microstates. This division of energy is usually called ‘being at the same temperature’ and so one can identify $\frac{\partial \Omega(E)}{\partial E}$ with the temperature $T$ (so that $T_1 = T_2$).

The temperature $T$ can now be defined by
$$\frac{1}{k_BT}=\frac{\partial \Omega(E)}{\partial E}.$$

Yes, what you derived becomes the definition of the thermal equilibrium in statistical thermodynamics when thermodynamic limit is taken into account because in that case we are basically certain to find internal energy of the system being close to the value which maximizes the total number of microstates.

Thank you for writing this out.

 

[Dario56]

On basis of a notification by @bob012345, one should specify what is meant by expressions like ‘the temperature is fixed” or ‘at a given temperature’ in context with the canonical ensemble approach. If a system $S_1$ is thermally coupled to a system $S_2$, one has for the number of microstates which can realize a configuration with $E_1$ and $E_2=E_t-E_1$
$$\Omega(E_t)=\Omega_1(E_1) \Omega_2(E_2)=\Omega_1(E_1) \Omega_2(E_t-E_1),$$
where $E_t=E_1+E_2$ is the total energy of the compound system.

The most likely division of energy between $S_1$ and $S_2$ is the one that maximizes $\Omega_1(E_1) \Omega_2(E_2)$ because this will correspond to the greatest number of possible microstates. For the most likely division of energy between the systems in thermal contact one has the condition
$$\frac{\partial \Omega_1(E_1)}{\partial E_1} = \frac{\partial \Omega_2(E_2)}{\partial E_2}$$
since it maximizes the total number of microstates. This division of energy is usually called ‘being at the same temperature’ and so one can identify $\frac{\partial \Omega(E)}{\partial E}$ with the temperature $T$ (so that $T_1 = T_2$).

The temperature $T$ can now be defined by
$$\frac{1}{k_BT}=\frac{\partial \Omega(E)}{\partial E}.$$

There is one thing which I am not sure about. As you pointed out, the canonical ensemble is defined for the system being in thermal equilibrium with the reservoir.

We established that the concept of thermal equilibrium really rather makes sense in the thermodynamic limit because in that case we are certain to find our system with the internal energy very close to its maximizing value with respect to the total number of microstates.

However, shouldn't the canonical ensemble apply in the more general case (not only in the thermodynamic limit)? After all, we use the canonical ensemble to derive probability distribution for energy states of the system. It is than that we apply the thermodynamic limit to find out properties of the most real systems where number of particles is huge.

In another words, how can the canonical ensemble be valid generally (not only in the thermodynamic limit), but use the restriction of thermal equilibrium (when deriving probability distribution of the energy states) which makes sense only in the thermodynamic limit?

 

[Lord Jestocost]

I have the feeling that the conception of 'ensembles' in statistical physics needs some clarification. I refer to the book “Statistical Mechanics of Solids” by Louis A. Girifalco:

“Imagine an enormous number of systems all of which are replicas of the physical system under consideration and all subject to the same external conditions as the physical system. Now place imaginary walls around each system and stack them together to form one huge, continuous mass. Such a collection is called an ensemble, and each replica is a member system of the ensemble.
The nature of the boundaries between the systems determines the type of ensemble:
1. If the boundaries are impermeable and each system is completely isolated, there are no interactions among the systems and the ensemble is called microcanonical.
2. If the walls are permeable to energy but not to anything else, only energy can be transferred among systems and the ensemble is called canonical.
3. If both energy and matter can be exchanged among systems, but the walls are impermeable to all other influences, the ensemble is called grand canonical.
…..
The utility of the concept of ensembles lies in the fact that averages of physical quantities can be obtained by averaging over the member systems of the ensemble. These ensemble averages are then identified with experimentally observed properties. In essence, the ensemble averages are equated to the time averages defined in the preceding section. This sidesteps the need to compute time averages and provides a consistent scheme to arrive at a statistical theory of macroscopic properties. The equality of time and ensemble averages is called the ergodic hypothesis, and its ultimate justification lies in the agreement of the results of statistical mechanics with experiment.

Clearly, in all ensembles except microcanonical ones, fluctuations will occur in at least some values of the physical variables of the member systems. Essentially, what is done in constructing an ensemble is to replace fluctuations caused by external influences, by fluctuations caused by internal interactions among various parts of the system. In so doing, the ensemble is taken to be a supersystem isolated from the rest of the world. Different kinds of ensembles consider fluctuations of certain properties and neglect others. For example, in the pressure grand canonical ensemble, fluctuations in volume, number of atoms, and energy are all taken into account, while in a canonical ensemble only energy fluctuations are considered.”
[bold by LJ]

 

[Dario56]

I have the feeling that the conception of 'ensembles' in statistical physics needs some clarification. I refer to the book “Statistical Mechanics of Solids” by Louis A. Girifalco:

“Imagine an enormous number of systems all of which are replicas of the physical system under consideration and all subject to the same external conditions as the physical system. Now place imaginary walls around each system and stack them together to form one huge, continuous mass. Such a collection is called an ensemble, and each replica is a member system of the ensemble.
The nature of the boundaries between the systems determines the type of ensemble:
1. If the boundaries are impermeable and each system is completely isolated, there are no interactions among the systems and the ensemble is called microcanonical.
2. If the walls are permeable to energy but not to anything else, only energy can be transferred among systems and the ensemble is called canonical.
3. If both energy and matter can be exchanged among systems, but the walls are impermeable to all other influences, the ensemble is called grand canonical.
…..
The utility of the concept of ensembles lies in the fact that averages of physical quantities can be obtained by averaging over the member systems of the ensemble. These ensemble averages are then identified with experimentally observed properties. In essence, the ensemble averages are equated to the time averages defined in the preceding section. This sidesteps the need to compute time averages and provides a consistent scheme to arrive at a statistical theory of macroscopic properties. The equality of time and ensemble averages is called the ergodic hypothesis, and its ultimate justification lies in the agreement of the results of statistical mechanics with experiment.

Clearly, in all ensembles except microcanonical ones, fluctuations will occur in at least some values of the physical variables of the member systems. Essentially, what is done in constructing an ensemble is to replace fluctuations caused by external influences, by fluctuations caused by internal interactions among various parts of the system. In so doing, the ensemble is taken to be a supersystem isolated from the rest of the world. Different kinds of ensembles consider fluctuations of certain properties and neglect others. For example, in the pressure grand canonical ensemble, fluctuations in volume, number of atoms, and energy are all taken into account, while in a canonical ensemble only energy fluctuations are considered.”
[bold by LJ]

Yes, thank you for the detailed explanation. I think I understand concept of ensemble in statistical thermodynamics well. As mentioned, in the derivation of the canonical ensemble, we defined thermodynamic variables (internal energy for example) as the expectation value over the whole ensemble.

For temperature however, we defined it differently. You derived in the previous comments what the concept of thermal equilibrium means in the context of statistical thermodynamics. This is important as the canonical ensemble is defined in the condition of thermal equilibrium.

This definition isn't the expectation value over the ensemble, but it has to do how number of microstates of the system and reservoir change with their interal energies.