Statistical Foundation of Thermodynamics

P.K. Ahluwalia

1. Learning Outcomes

After studying this module, you shall be able to:

• Understand the meaning of thermodynamic limit and importance of particle density.
• Differentiate between macrostate and microstate of a system.
• Understand the notion of equal likely possibility of each of the microstates of a given macrostate, principle of equal apriori probability, as the corner stone of statistical mechanics.
• Count number of possible microstates for a given macrostate leading to idea of thermodynamic probability.
• Establish the meaning of statistical mechanical temperature, pressure and chemical potential and show its equivalence with thermodynamic temperature, pressure and chemical potential.
• Define statistically the Boltzmann formula for entropy.
• Have a recipe for deriving thermodynamics from statistical route.

2. Introduction

In this module from the inherent statistical nature of macroscopic systems, we underline the statistical formulation of thermodynamics. We will see how this approach provided microscopic basis to thermodynamics by successful application of this approach linking entropy with the probability of microstates overcoming the lack of information about the
system made up of a large number of particles of the order of avogadro’s number. We shall see that the statistical-mechanic meaning of temperature, pressure and chemical potential is the same as thermodynamic meaning of temperature, pressure and chemical potential. Las of all we shall later try to get a prescription to derive thermodynamic functions related to the system under study. System of study shall always be treated as a system in equilibrium.

3. Thermodynamic Limit

Realizing the fact that macroscopic systems are made up of typically ?. ??? × ???? points to the enormity of the systems involved. Thermodynamic limit is a way to express this enormity and a natural way to express this by carrying out the limit number of particles ? → ∞ and volume ? → ∞, such that the particle density ? remains fixed. Taking this limit amounts to saying that extensive properties of the system varies proportional to the size of the system i.e. ? or ? where as intensive properties remain independent of the size of the system. The process can be start with a finite system with a fixed density and keep on adding more and more such identical finite systems to it such that number density remains the same causing increase only in extensive parameters without change in the intensive parameters. Not going to thermodynamic limit in the mathematical sense can lead to serious problems with the strict compliance of the extensive nature of extensive parameters such as entropy.

4. Macroscopic State and Microscopic State

There are two ways to characterize a macroscopic system in equilibrium under consideration. These two ways are respectively macroscopic state and microscopic state of the system. A given macroscopic state corresponds to a large number of different microstates.

A macroscopic state also called a macrostate is described in terms of macroscopic variables such as total energy ?, volume ? and number of particles ?. In describing a macrostate we are not at all bothered by the description of the properties of individual particles constituting the system such as their momentum, spin etc.

A microscopic state also called microstate is specified by describing the state of each particle in a given macrostate. Counting the number of microstates in a given macrostate is the most basic task to indulge in statistical mechanics.

To further appreciate the difference, let us take a system m ade up of N number of non-interacting particles with total energy ?. This energy is the sum of the energies ?? of individual particles, such that

? = Σ????                                                                 (1)
?

Where ?? is the number of particles each with energy ?? . Obviously.

? = Σ??                                                                    (2)
?

It may be recalled (see appendices A1 and A2) that ??are the single particle energies which have two distinct properties:

(i) These are quantized, leading to quantization of total energy ?

(ii) Their value depends on the volume to which they are confined. However, it is worth noting that as volume increases the difference between the adjacent energy levels decrease rapidly and E goes from being a discreet variable to a continuous variable.

So macrostate of a system under consideration is defined by specification of ?, ? and ?. This macrostate can be realized at the microscopic level in many different possible ways known as microstates. In simple words, it implies that total energy ? can be distributed among ? particles constituting the system in large numer of different ways. Quantum mechanically this amounts to specifying all possible independent solutions ?(??, ??, ??, … … … . ??) of the N particle Schrodinger equation of the system under consideration with energy eigen value ?.

One of the aims of statistical mechanical formulation is count the number of all the possible microstates. In this course this number shall be denoted by ?, which is a function of macroscopic variables of the system, which in the present case are ?, ? and ?. Therefore, this count is a number denoted by the function ?(?, ?, ?). The value ?/?, gives the pro bability
of ocuurance of any one of the states, which is as per the principle of equal apriori probability the same for each microstate. This ?/? , is also known as thermodynamic probability. ??.
The equilibrium state of the system then is the state for which function ?(?, ?, ?) maximizes for each of its arguments. It is interesting to note that though a system corresponding to agiven macrostate remains in a state of equilibrium, its microscopic states may keep on changing continuously.

The equilibrium state of the system then is the state for which function ?(?, ?, ?) maximizes for each of its arguments. It is interesting to note that though a system corresponding to a
given macrostate remains in a state of equilibrium, its microscopic states may keep on changing continuously.

Principle of Equal Apriori Probabilities

This principle is the back bone of the statistical mechanics for systems at equilibrium. It states that all allowed microstates compatible with a given macrostate, (i.e. subject to constraints imposed by equations (1) and (2) or any other constaint on macroscopic variables involved) are equally likely.

Allowing possibility of all microstates of a given macrostate has resulted in derivation of ideal gas laws and derivation of Maxwell Boltzmann velocity distribution law which were later
confirmed experimentally. Justification of this principle, therefore, lies in the results derived from it and observed experimentally. This principle is treated as one of the postulates of statistical mechanics.

6. Linking Thermodynamics with Statistics

We now proceed to establish a link between thermodynamics with statistics based on the the state that the equilibrium state of the system then is the state for which function ?(?, ?, ?) maximizes for each of its arguments leading to three conditions of equilibrium for each of the microscopic variable ?, ? and ?. For this we consider two physical systems: system 1 and system 2 which are separately in equilibrium, described by the macroscopic variables ??, ??, ?? and ??. ??, ??. System 1 has ??(??, ??, ??) number of microstates and system 2 has ??(??, ??, ?) as shown in figure

_{Figure 1 Two systems ready for making contact in three ways (a) when the partition is conductiong (b) when the partition is made conducting as well as movable and (c) when the partition also allows for exchange of particles.}

Case I: Thermal Contact Between Two Systems: Notion of Temperature in Statistical Mechanics: Let us consider first the case in which ??, ??and ?? ?? are kept constant but energy can be exchanged by making the intervening partition conducting. However, in this case the total energy ? = ?? + ?? remains conserved i.e.

? = ?? + ?? = ????????                                                 (3)

At any instant system 1 is equally likely to be in any one of the microstates  ??(??, ??, ??) and system 2 is equally likely to be in any one of the microstates ??(??, ??, ??). Then the combined system is equally likely to be any one of the microstates ? (?, ?, ?), such that

?(?, ?, ?) = ??(??, ??, ??) ?(??, ??, ??)
= ?(??, ??, ??)?(??, ??, ? − ??)                                 (4)

Since ??, ??and ??, ??are constant, we can write (4) as

 ?(?) = ?( ??)?(? − ??)                                                  (5)

? number of microstates of the combined system here vary with ??till equilibrium is established, and this happens until ? maximizes. The combined system then attains a macrostate which has the most number of microstates. The important point to note is that this macrostate of the system is that state in which the system spends the largest portion of its time and presents itself as an equilibrium state.

Mathematical this condition when applied to (5) means

What have we achieved? We found that

Case II: Mechanical Contact Between Two Systems: Notion of Pressure in Statistical Mechanics :

If we extend the considerations of the case I and further relax the constraint on the partition separating system 1 from system 2 by making it movable as well as conducting figure 1(b), then the volumes of the two systems also become variables, though the total volume? = ?? + ??, remains constant and besides equation (7) corresponding to thermal equilibrium.

Case III: Chermical Contact Between Two Systems: Notion of Chemical Potential in Statistical Mechanics :

If we combine the three case discussed above such that if we allow exchange of energy, change of volume and exchange of particles simultaneously, the conditions of equilibrium are

                       ?? = ??, ?? = ?? and ?? = ??                                         (23)

7. Recipe for Deriving Thermodynamics from Statistical Route

9. Summary

In this module we have learnt

About statistical formulation of thermodynamics of a system in equilibrium by knowing the accessible microstates of a macrostate of the given system.

• That as per principle of equal apriori probability, each of the accessible microstates is equally likely to happen.
• That thermodynamic limit means ? → ∞ and ? → ∞ keeping the number density ?/? constant.
• That entropy of a given macrostate of a system is related to the number of allowed microstates ?(?, ?, ?) by the famous Boltzmann formula

? = ?? ?? ?

• That equilibrium state of the system is that state for which ?(?, ?, ?) maximizes, which further means maximization of entropy ?.
• That the equilibrium conditions for two systems under different types of contacts are

• That the knowledge of number of accessible microstates provides complete knowledge of the equilibrium thermodynamics of the system via Boltzmann formula.

References:-

1. Pathria R.K. and Beale P. D., Statistical Mechanics, 3rd ed. (Elsevier, 2011).

2. Landau L.D., Lifshitz E.M., “ Statistical Physics Part 1,” 3rd Edition, Oxford: Pergamon Press., 1982.

3. Pal P.B., “An Introductory Course of Statistical Mechanics”, New Delhi: Narosa Publishing House Pvt. Ltd., 2008.

4. Panat P.V., “Thermodynamics and Statistical Mechanics,” New Delhi: Narosa Publishing House Pvt. Ltd., 2008.

5. Ma S.K., “ Statistical Mechanics,” Singapore: World Scientific Publishing Co. Pte. Ltd., 1985.

6. Greiner W., Neise L., Stocker H., “ Thermodynamics and Statistical Mechanics,” New York, Springer Verlag, 1995.

Appendices

Following appendices provide a recapitulation of how quantization of energy levels arises and how the value of single particle energy in a confined system is volume dependent.
We shall try to elaborate on the concept of degeneracy which is auseful idea in counting the microstates of a physical system.

A1 Particle in a one dimensional box:

We are interested in quantum mechanical description of a particle confined to move on a straight line bounded by two walls at each end at a distance ?. The particles make elastic collisions at the two walls. There is no force acting on the particle.