Thermo Postulates of Callen and Robertson, etc.

 (see my previous post for a general comparison of the C and R textbooks)

Callen:

I.    There exist particular states (called equilibrium states) of simple systems that, macroscopically, are characterized completely by the internal energy U, the volume V, and the mole numbers N₁, N₂, …, N_r of the chemical components.

II.   There exists a function (called the entropy S) of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states.

III.  The entropy of a composite system is additive over the constituent subsystems. The entropy is continuous and differentiable and is a monotonically increasing function of the energy. [Callen then adds: Several mathematical consequences follow immediately.]

IV.  The entropy of any system vanishes in the state for which (∂U/∂S) = 0, with volume and all chemical potentials held constant. That is, the entropy goes to zero at the zero of absolute temperature.

 

Robertson:

1.  The macroscopic thermodynamic equilibrium states of simple systems are completely specified in terms of the extensive parameters (U, V, N_i in the present context), where the N_i are the component populations, with all the information contained in the entropy function S(U,V,N_i), called the fundamental relation.

2.  The entropy of a simple system is a continuous, differentiable, homogeneous-first-order function of the extensive parameters, monotone increasing in U.

3.  The entropy of a composite system S({U_i, V_i, {N_i}}) is the sum of the entropies of the constituent subsystems:

S({U_i, V_i, {N_i}}) = S₁({U₁, V₁, N_1i }) + S₁({U₂, V₂, N_2i }) + … .

4.  The equilibrium state of a composite system when a constraint is removed maximizes the total entropy over the set of all possible constrained equilibrium states compatible with the constraint and the permitted range of extensive parameters.

5.  The entropy of a simple system approaches zero when

(∂U/∂S)_{V, {N}} → 0

 --------

Like Callen, Robertson uses the idea of equilibrium states of simple macroscopic systems as a starting point, with equilibrium sort of implicitly taken to be any state that can be characterized completely by a constant internal (average) energy, constant volume, and constant “component populations” (Robertson) or “mole numbers of the chemical components” (Callen).

Notice that C and R both use a completeness specification. The word “completeness” resonates a little bit here with the “complete set of commuting observables” in quantum mechanics. We could say an equilibrium state in thermodynamics is characterized by—or exists because of— the existence of stationary values of a complete set of extensive parameters,  which are U, V and {N_i} in the entropy representation.

Also notice that a fundamental relation such as the monotonically increasing entropy function S(U,V,N_i) in thermodynamics is described similarly to the wave function in quantum mechanics, in that both are said to contain all the information about the system.

An aside related to vocabulary

The word “system” should be used with some humility and caution, rather like the word “universe”. An ideal isolated system in thermodynamics is a universe unto itself (if you don’t tamper with it), while, conversely, the universe is a system unto itself.  What are they really? Models. Mainly, “system” is a very broadly used word in science and engineering and it can close off creative thinking rather than promoting it. Some people—Darwin and Fowler in their 1922 and 1923 papers, and Schrödinger in his little Statistical Thermodynamics book—have chosen to use the word “assembly” instead of “system” when discussing Boltzmann’s ideal gas and Planck’s ideal electromagnetic resonators. These authors also use the word “system,” but they refer to the individual molecules or Planck resonators/vibrators/oscillators as the systems that make up the assembly under consideration.  Thus, in their view, an assembly is macroscopic and must be assembled, and its “component population” is made of N identical (sub)microscopic systems that each possess mechanical and maybe electromagnetic energy (KE, PE). The assembly itself then has some overall thermal energy distribution. A more complicated assembly would be made up of a set {N_i} of different types of systems.

Now back to (thermodynamic) systems analysis

But I will continue talking about thermodynamic “systems” and their constituents since this is the usual terminology.

Before he provides the above postulates (Chapter 2, p. 66), Robertson describes a simple system (his bold emphasis) as “a bounded region of space that is macroscopically homogeneous.” He goes on to say: “That we regard a system as simple may imply nothing more than that we have not examined it on a fine enough scale. The failure to do so may be one of choice, or lack of it. We may usually choose to regard an obviously complex system as simple if its internal behavior is not involved in the problem at hand … the simple systems that are treated in terms of statistical thermophysics are made up of atoms or molecules, the spatial distribution of which is described by probability densities that are constant (or periodic on the scale of a crystal lattice) over the volume of the system.” Robertson then discusses the nature of possible boundaries of simple systems, such as their being either material or “described by a set of mathematical surfaces in space,” or diathermal (allowing thermal contact) or adiabatic (preventing thermal contact), or restrictive to matter flow in various degrees (semipermeable, open, closed), and whether they allow transfer of energy via a work process (such as a movable piston).

I’ve discussed Robertson’s and Callen’s statements of the postulates of thermodynamics in this post in order to prepare for my next post, where I’ll compare these postulates with those of quantum mechanics and also, mainly, try to figure out why we don’t normally see the square of the wavefunction or the squares of the complex quantum mechanical superposition coefficients used as probabilities in the Shannon expression for entropy. Meanwhile, here’s a blog post on that subject: Wavefunction entropy. [The comparison of thermo and quantum postulates wasn't my next post. As of December 16, 2023, I still haven't managed to get to it. Later!]

 

Postscript, March 22: Callen's postulate II and Robertson's corresponding postulate (No. 4) are too abstract to be understood, at least for me, without an example.  Callen gives a general sort of example, and also gives a problem at the end of the section (1.10) to further illustrate the example.

But first let's consider the problem that these particular postulates of C and R are supposed to solve. It's el problema grande of thermodynamics, which as Callen states it (and I've previously quoted) is: The single, all-encompassing problem of thermodynamics is the determination of the equilibrium state that eventually results after the removal of internal constraints in a closed, composite system.

Example: (page 26) "Given two or more simple systems, they may be considered as constituting a single composite system. The composite system is termed closed if it is surrounded by a wall that is restrictive with respect to the total energy, the total volume, and the total mole numbers of each component of the composite system." ... (page 28) "In the case of two systems separated by a diathermal wall we might wish to predict the manner in which the total energy U distributes between the two systems. We then consider the composite system with the internal diathermal wall replaced by an adiabatic wall and with particular values of U⁽¹⁾ and U⁽²⁾, with U⁽¹⁾ + U⁽²⁾ = U. For each such constrained equilibrium state there is an entropy of the composite system, and for some particular values of U⁽¹⁾ and U⁽²⁾ this entropy is a maximum. These, then, are the values of U⁽¹⁾ and U⁽²⁾ that obtain in the presence of the diathermal wall, or in the absence of the adiabatic constraint."

Problem 1.10-3: "The fundamental equation of system A is

S = (R²/v₀ θ)^1/3(NVU)^1/3

and similarly for System B. The two systems are separated by a rigid, impermeable, adiabatic wall. System A has a volume of 9x10^-6 m³ and a mole number of 3 moles. System B has a volume of 4x10^-6 m³ and a mole number of 2 moles. The total energy of the composite system is 80 J. Calculate and Plot the entropy as a function of U_A/(U_A + U_B) . If the internal wall is now made diathermal and the system is allowed to come to equilibrium, what are the internal energies of the individual systems? (R², v₀ , and θ are constants.)"

Post-postscript, March 25: (The red text above is what I left out or wrote wrongly in my initial post. The red text below is what I re-wrote on March 29.) Non-numerical solution to our Problem 1.10-3: The given constraint U = U_A + U_B applies to the composite system with either the adiabatic wall or the diathermal wall. The composite system entropy sum S = S_A + S_B applies when the adiabatic wall is in place and subsystems A and B are energetically distinct, AND when the diathermal wall is in place with the particular values of U_A and U_B found from maximizing S = S_A + S_B.  These are the thermal equilibrium values with the diathermal wall in place.

We have a continuum of different values for S_A and S_B that satisfy the sum S = S_A + S_B with the adiabatic wall in place, and these are Callen’s and Robertson’s “constrained equilibrium states” over which we want to maximize S. Using the energy constraint to write total entropy in terms of system A’s energy, and using constants k_A and k_B as stand-ins for all the alphabetic and numerical constants given in the problem,

S  =  S_A + S_B   =  k_A U_A ^1/3 + k_B U_B ^1/3

=  k_A U_A ^1/3 + k_B (U – U_A)^1/3

dS/dU_A =  (k_A /3) U_A ^-2/3 –  (k_B /3)(U – U_A)^-2/3  = 0,

(not checked yet for min instead of max) resulting in

U_A = U/[1 +  (k_B / k_A)^3/2]

and

U_B = U/[1 +  (k_A / k_B )^3/2].

The ratios are easy to calculate, with alphabetic constants and numerical exponents canceling:  k_A / k_B  = 27/8.  Plotting the normalized relation "entropy as a function of U_A/(U_A + U_B)" is left to the intrepid reader for the moment.