The Canonical Ensemble

2-1 Ensemble Averages

Calculate thermodynamic properties in terms of molecular properties. Do first with respect to mechanical properties, which are quantum mechanical or classical mechanical quantities, and then nonmechanical thermodynamic variables. A useful distinction between mechanical and nonmechanical properties is that mechanical properties are defined without appealing to the concept of temperature, whereas the definitions of nonmechanical properties involve temperature.

Consider some macroscopic system of interest. There are an enormous number of quantum states consistent with the fixed macroscopic properties. In order to calculate the value of any mechanical property, one calculates the value of that mechanical propertiy in each and every one of the quantum states that is consistent with the few parameters necessary. The average of these mechanical properties is then taken, giving each possible quantum state the same weight.

An ensemble is a virtual collection of a very large number of systems each constructed to be a replica on a thermodynamic level of the particular thermodynamic system of interest. Consider an isolated system with <tex>N</tex>, <tex>V</tex>, and <tex>E</tex> fixed. The corresponding values of the ensemble would be the value of the property multiplied by the number of systems. The values of <tex>N</tex> and <tex>V</tex> along with the force law between the molecules are sufficient to determine the energy eigenvalues <tex>E_j</tex> of the Schrodinger equation along with their associated degeneracies <tex>\Omega (E_j)</tex>.

An interpretation of the principle of equal a priori probabilities is that an isolated system (<tex>N</tex>, <tex>V</tex>, and <tex>E</tex> fixed) is equally likely to be in any of its <tex>\Omega (E)</tex> possible quantum states.

Define ane ensemble average of a mechanical property as the average value of the probability over all members of the ensemble, utilizing the principle of equal a priori probabilities.

Ignore the uncertainty in energy. A quantum mechanical system will, in general, not be in one of the <tex>\Omega (E)</tex> selected states, but will be some linear combination of them.

A goal is to caclulate the ensemble average of some mechanical property and then show that this can be set equal to the corresponding thermodynamic property.

Ensembles whose members are with fixed <tex>N</tex>, <tex>V</tex>, and <tex>E</tex> are called microcanonical ensembles. The most commonly used ensemble in statistical mechanics is the canonical ensemble, in which the ensembles are with fixed <tex>N</tex>, <tex>V</tex>, and <tex>T</tex>.

2-2 Method of the Most Probable Distribution

Consider an experimental system with <tex>N</tex>, <tex>V</tex>, and <tex>T</tex> as independent variables. Mentally construct an ensemble of systems . Enclose each system in a container of volume <tex>V</tex> with walls that are heat conducting but impermeable to the passage of molecules. Place the entire ensemble of systems in a very large heat bath at temperature <tex>T</tex>. The energy of each system is not fixed at any set value. Consider the entire spectrum of energy states for each member of the canonical ensemble. Any particular system might be found in any of the quantum states. The average energy of the probability that some system has a certain energy depends on the temperature. Any set of energies is possible and must be considered. The occupation numbers satisfy two conditions.

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Apply the principle of equal a priori probabilities to this isolated system. The number of ways that any particular distribution of the <tex>a_j</tex>'s can be realized is the number of ways <tex>A</tex> distinguishable objects can be arranged into groups, such that <tex>a_1</tex> are in the first group, <tex>a_2</tex> in the second, and so on.

<tex>W(\vec a ) = \frac

Unknown macro: {A!}

Unknown macro: {a_1! a_2! a_3!...}

</tex>

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The overall probability <tex>P_j</tex> that a system is in the <tex>j</tex>th quantum state is obtained by averaging <tex>a_j / A</tex> over all the allowed distributions, giving equal weight to each one according to the principle of equal a priori probabilities.

<tex>P_j = \frac

Unknown macro: {bar a_j}

Unknown macro: {A}

</tex>

<tex>P_j = \frac

Unknown macro: {1}

\frac{\sum_{\vec{a}} W ( \vec a ) a_j ( \vec

Unknown macro: {a}

)}{\sum_

Unknown macro: {vec a}

W(\vec a )}</tex>

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Calculate the canonical ensemble average of any mechanical property from the relation below.

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The distribution at which <tex>W(\vec a)</tex> is a maximum is <tex>\vec a^* =

Unknown macro: {a_j^*}

</tex>. The <tex>W(\vec a )</tex> at any set of <tex>a_j</tex>'s other than the set \vec a* are completely negligible.

<tex>P_j = \frac

Unknown macro: {1}

Unknown macro: {A}

\frac{W(\vec

^)a_j}{W(\vec

Unknown macro: {a}

^*)}</tex>

<tex>P_j = \frac

Unknown macro: {vec a_j^*}

Unknown macro: {A}

</tex>

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The term <tex>a_j^</tex>* is the value of <tex>a_j</tex> in that distribution that maximizes <tex>W(\vec a)</tex>. To calculate the probabilities to be used in ensemble average, determine only the distribution <tex>\vec a^</tex> that maximizes <tex>W(a^)</tex> under two constraints. Solve using Lagrange's method of undetermined multipliers. Refer to class notes.

2-3 The Evaluation of the Undetermined Multipliers, alpha and beta

Refer to class notes to find expression of thermodynamic energy and pressure.

The term <tex>k</tex> is the same value for all substances. Consider two closed systems <tex>A</tex> and <tex>B</tex> each with its own kind of particles and energy states, but in thermal contact with each other and immersed in a heat bath of temperature <tex>T</tex>. Construct a canonical ensemble of systems <tex>AB</tex> representative of a thermodynamic <tex>AB</tex> system at temperature <tex>T</tex> and apply the method of the most probable distribution to the <tex>AB</tex> system.

<tex>W( \vec a, \vec b ) = \frac

Unknown macro: {Pi_j a_j!}

\cdot \frac

Unknown macro: {B!}

Unknown macro: {Pi_k b_k!}

</tex>

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There are three relations that must be satisfied.

<tex>\sum_j ( a_j E_

Unknown macro: {jA}

+ b_j E_

Unknown macro: {jB}

= E</tex>

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Apply the method of the most probable distribution for the simultaneous probability that the <tex>AB</tex> system includes <tex>A</tex> in the <tex>i</tex>th quantum state and <tex>B</tex> in the <tex>j</tex>th quantum state.

<tex>P_

Unknown macro: {ij}

= \frac{e^{-\beta E_

Unknown macro: {iA}

}}

Unknown macro: {Q_A}

\frac{e^{-\beta E_

}}

Unknown macro: {Q_B}

</tex>

<tex>P_

Unknown macro: {ij}

= P_

Unknown macro: {iA}

P_

Unknown macro: {jB}

</tex>

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The terms of <tex>Q</tex> are the partition function of the respective systems. The term <tex>k</tex> is a universal constant and can be evaluated using any convenient system.

Show that <tex>\beta dq_

Unknown macro: {rev}

</tex> is an exact differential. Consider the function <tex>f = \ln Q</tex>. Write the differential in the equation below.

<tex>d \left ( f + \beta \bar E \right ) = \beta \left ( d \bar E - \sum_j P_j dE_j \right )</tex>

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Change the volume of all the systems by <tex>dV</tex>, changing the <tex>E_j</tex>'s for all of them alike in order that there is still an ensemble of macroscopically identical systems. Also change the temperature of the ensemble by <tex>dT</tex> by coupling it with a large heat bath, changing the temperature slightly and then isolating the ensemble from the heat bath.

<tex>d \bar E - \sum_j P_j dE_j = \beta \delta q_

</tex>

<tex>S = \frac

Unknown macro: { bar E }

Unknown macro: {T}

+ k \ln Q + \mbox

Unknown macro: {constant}

</tex>

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Work is done by changing energies slightly and keeping the population of the states fixed. When a small quantity of heat is absorbed from the surroundings, the energy states of the system do not change, but the population of these states do.

2-4 Thermodynamic Connection

Write <tex>E</tex>, <tex>p</tex>, and <tex>S</tex> as functions of <tex>Q</tex>.

<tex>\bar E = kT^2 \left ( \frac

Unknown macro: { partial ln Q }

Unknown macro: { partial T}

\right )_

Unknown macro: {N, V}

</tex>

<tex>\bar p = kT \left ( \frac

Unknown macro: {partial ln Q}

Unknown macro: {partial V}

\right )_

Unknown macro: {N, T}

</tex>

<tex>S = kT \left ( \frac

Unknown macro: {partial ln Q }

Unknown macro: { partial T }

\right )_

+ k \ln Q</tex>

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The partition function is a bridge betweent the quantum mechanical energy states of a macroscopic system and the thermodynamic properties of that system. The system is often approximated by classical mechanics.

Derive an equation for the Hemholtz free energy <tex>A</tex> in terms of <tex>Q</tex>.

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The above equation can be considered the most important connection between thermodynamics and the canonical partition function, since it is possible to derive many equations starting with its differential form.

A statement of the second law of thermodynamics for closed, isothermal systems is that <tex>\Delta A < 0</tex> for a spontaneous process. Below is an expression that is useful when summing over the levels of a system.

<tex>Q(N, V, T_ = \sum_E \Omega (N, V, E) e^{-E(N, V)/kT}</tex>

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Consider a typical spontaneous process. Refer to example in notes about expanding gas. The number of states to an expanding gas increases. The addition of a catalyst can remove the constraint of a high activation energy barrier, and this results in an increase in the number of accesible states. The removal of some constraints allows a greater number of quantum states to be accessible to a system, and the "flow" of the system into states is observed as a spontaneous process.

Consider an isothermal process. When a system is in a heat bath rather than isolated, we must include all possible energy states or levels of the system.

<tex>Q_2 - Q_1 = \sum_E

Unknown macro: {Omega_2 (N, V, E) - Omega_1 (N, V, E)}

e^{-E/kT} > 0</tex>

<tex>\Delta A = A_2 - A_1</tex>

<tex>\Delta A = -kT \ln \frac

Unknown macro: {Q_2}

Unknown macro: {Q_1}

</tex>

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The second law of thermodynamics has been written.

Consider the implications of putting the "constant" equal to zero.

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As the temperature approaches zero, entropy is proportional to the logarithm of the degeneracy of the lowest level.

Child pages

McQuarrie Chapter 2

2-1 Ensemble Averages

2-2 Method of the Most Probable Distribution

2-3 The Evaluation of the Undetermined Multipliers, alpha and beta

2-4 Thermodynamic Connection