On Classical Gases.

# On Classical Gases.

Jacques Arnaud
Mas Liron, F30440 Saint Martial, France
Laurent Chusseau
IES, UMR n5214 au CNRS, Université Montpellier II, F34095 Montpellier, France
Fabrice Philippe
LIRMM, UMR n5506 au CNRS, 161 rue Ada, F34392 Montpellier, France
July 2, 2019
###### Abstract

The ideal gas laws are derived from the democritian concept of corpuscles moving in vacuum plus a principle of simplicity, namely that these laws are independent of the laws of motion aside from the law of energy conservation. A single corpuscle in contact with a heat bath and submitted to a and -invariant force is considered, in which case corpuscle distinguishability is irrelevant. The non-relativistic approximation is made only in examples. Some of the end results are known but the method appears to be novel. The mathematics being elementary the present paper should facilitate the understanding of the ideal-gas law and more generally of classical thermodynamics. It supplements importantly a previously published paper: The stability of ideal gases is proven from the expressions obtained for the force exerted by the corpuscle on the two end pistons of a cylinder, and the internal energy. We evaluate the entropy increase that occurs when the wall separating two cylinders is removed and show that the entropy remains the same when the separation is restored. The entropy increment may be defined at the ratio of heat entering into the system and temperature when the number of corpuscles (0 or 1) is fixed. In general the entropy is defined as the average value of where denotes the probability of a given state. Generalization to -dependent weights, or equivalently to arbitrary static potentials, is made.

## 1 Introduction

This paper gives an alternative derivation of the classical barometric and ideal gas laws. In the title the word “classical” means: “non-quantum” (). Our results coincide with those obtained from the Bohr-Sommerfeld (BS) quasi-classical approximation of Quantum Mechanics and the Boltzmann factor but the method is more straightforward. Initially we obtained the average force exerted by a corpuscle on a piston from the BS theory. The corpuscle action (area in the position-momentum - phase-space, where denotes the corpuscle altitude and the momentum , where is the force applied to the corpuscle, e.g., its weight, and is time) is discrete, and evenly spaced in units of the Planck constant . Because the “bouncing ball” presently considered is not an harmonic oscillator the discrete corpuscle energies are not evenly spaced. This is why, when going to the continuous limit converting the sum into an integral we must introduce a distribution in the form given later. After going through these semi-classical considerations we discovered that it was sufficient to postulate the simplicity principle according to which the average force must not depend on the equations of motion. This concept, unlike the quantum theory, could have been understood at the time of the ancient Greece. Going the opposite way, one may say that the simplicity principle just stated suggests the semi-classical quantum theory. The entropy should be defined in general as the average value of where denotes the probability of a state, which accounts for a possible uncertainty concerning the presence of a corpuscle.

By ideal gas, we mean a collection of non-interacting corpuscles. We may therefore restrict ourselves to a single corpuscle so that considerations of corpuscles distinguishability are irrelevant. We suppose that we know with certainty whether a corpuscle is present or not, except in Section 8. Photons whose number may vary do not fulfill our definition of an ideal gas even though they are non-interacting; thus they are not treated.

Only motion along the vertical axis is considered, but generalization of the barometric and ideal-gas law to three dimensions is straightforward. As said earlier, the corpuscle is submitted to an external force , perhaps of electrical origin, being called the corpuscle “weight”, or equivalently to a potential . Because the presence of a corpuscle affects negligibly the potential the latter is an external potential. We call perfect gas ( “gaz parfait” in french, see[1]. We quote: “Le désordre initial de Démocrite peut faire songer au chaos moléculaire de Boltzmann qui, dans sa théorie des gaz parfaits, admettait comme une simple évidence que dans la situation de départ positions et vitesses des molécules sont réparties au hasard, avec l’hypothèse générale d’indépendance de ces paramètres deux à deux, indépendance que l’on trouve aussi chez Démocrite”) an ideal gas with no external force acting on the corpuscle except at the boundaries and with the non-relativistic approximation being made. In that case the constant-volume heat capacity is independent of volume and temperature. These definitions essentially agree with the ones given in[2]. We quote: “a distinction is made between an ideal gas, where the heat capacities could vary with temperature, and a perfect gas for which this is not the case”. In our one-dimensional model neglecting corpuscle rotations and vibrations the heat capacity (derivative of the gas internal energy with respect to temperature at fixed volume) is equal to 1/2. According to our definition an ideal gas is not necessarily a “Joule’s gas”: the heat capacity may depend on temperature and volume . The free expansion of an ideal gas may therefore entail temperature changes while this is not the case for a perfect gas. There is full agreement between our general results and the first and second laws of thermodynamics as they are spelled out in textbooks. There is also full agreement between our results and the usual perfect-gas laws that can be found in the Bernoulli work and elementary textbooks in the limit considered. We give here a generalized and simplified treatment that accounts for the effect of a constant force acting on the corpuscle and (among others) special-relativity effects. Some results in those cases are known, see Landsberg [3] and Louis-Martinez [4].

The present paper is a generalization of our previous papers[5, 6] where our motivation is explained in more detail than in the present paper. Presently, we calculate the average forces exerted on both ends of a vertical cylinder (they are different when the corpuscle has weight), discuss the gas stability, the entropy increments that occur when a separating wall is removed and restored, and consider weights that may vary with altitude, as is the case for example on earth for cylinder heights that are not negligible compared to the earth radius.

We first consider the round-trip time needed for a corpuscle thrown upward with energy is to reach an altitude above the ground level and come back to the ground level. If the corpuscle bounces elastically on the ground, represents the oscillation period. We consider only round-trip times, that is time delays measured at some altitude, so that no problem of clock synchronisation arises. The time during which the corpuscle is located above some altitude during a period is: , since, under our assumption of a constant weight the -function does not depend on the initial altitude or initial time. If the weight is of gravitational origin with acceleration , it is known from the Galileo experiment that the function does not depend on the corpuscle mass but this is not so in general.

We consider only thermal-equilibrium situations: If we wait a sufficiently long period of time an isolated system ceases to evolve. We take it as an empirical result that, leaving aside general-relativity effects111According to general relativity, thermal energy has weight. But this (so-called Tolman) effect that entails a temperature variation at equilibrium: , where denotes the gravity acceleration and the speed of light, is entirely negligible; see for example equation (1) of[7]. This paper gives the following interpretation of equilibrium: we quote “The temperature is essentially equal to divided by the time required by the system to move from one state to the next”. This interpretation leads to the condition of temperature uniformity for weak gravity since in that case time intervals do not depend significantly on altitude., two bodies left in contact for a sufficient period of time with energy being allowed to flow from one to the other, reach an equilibrium state corresponding to equal temperatures as one can judge by our senses. If energy may flow spontaneously (without work expenditure) from one body to the other, the converse is not possible: the process is non-reversible (zeroth law of thermodynamics).

The purpose of the present paper is thus to show that the thermodynamics of ideal gases and particularly the barometric and ideal-gas laws may be obtained on the sole basis of the Democritus model according to which nature consists of corpuscles moving in a vacuum, plus a principle of simplicity: namely that these fundamental laws are independent of the law of corpuscle motion: non-relativistic, special relativistic, or otherwise. They apply for example to deep-sea or capillary wave packets (see Appendix A). To wit, writing the corpuscle Hamiltonian as , the barometric and ideal-gas laws do not depend on the function.

In our discussion the temperature , where denotes the Boltzmann constant and the temperature in kelvin, enters solely for dimensional reasons. We later show that, remarkably, our expressions of the gas internal energy and force (or pressure) derive from the partial derivatives of the Helmholtz potential (or free energy) . Some of the subsequent calculations are conventional. The heat delivered by the gas is: , an expression for the entropy being given. This result enables us to prove that the formally-introduced temperature is a thermodynamic temperature. Indeed, we recover for ideal gases the general Carnot result asserting that the efficiency of reversible thermal engines is: , where denotes the cold bath temperature and the hot bath temperature. Since Kelvin time this expression defines the thermodynamic temperature to within an arbitrary proportionality factor, which is fixed by specifying that joules at the water triple point, that is: if we take joules as an energy unit. An alternative convention will be suggested.

As said earlier we consider a single corpuscle. Because of the slight thermal motion of the container wall there is an exchange of energy between the corpuscle and the heat bath so that the corpuscle energy slowly varies in the course of time. What we are looking for are averages over arbitrary long time intervals. We recall [8] that the fundamental Helmholtz relation: and the fundamental entropy relation: have the same mathematical and physical contents, being related by Legendre transforms. We verify that our expression for leads to stable equilibria. The usefulness of the entropy concept is of course that it may not decrease when constraints inside a thermally isolated object are removed or added (second law of thermodynamics). We will verify that this is so from our model formulas. Finally, we shall consider the case where the weight instead of being a constant varies by steps along the vertical axis, that is, in the limit, the case of arbitrary potentials .

The reader may feel that our statement that an invariance principle implies the barometric and ideal-gas laws without anything else is surprising. On the other hand a quick reading of standard books on Thermodynamics may lead other readers to believe that this is a well-known fact. For example, Callen [8] states correctly that: “The essence of the ideal-gas law is that molecules of the gas do not interact. This simple fact implies that ”. However, in order to reach this conclusion, that author needs postulate the Newtonian law of motion, quantum theory, and the Boltzmann factor. The claim that corpuscle independence entails the barometric and ideal-gas laws without (almost) anything else therefore does not appear to have been justified before.

## 2 The barometric law

We consider a single corpuscle moving only along the vertical coordinate and submitted to a constant force . If the corpuscle energy is , the maximum altitude reached is . The round-trip time is the motion period, and the time per period during which the corpuscle is above is . If follows that the fraction of time during which the corpuscle is above the -level is . To obtain an expression independent of the -function, one must introduce an energy distribution , and define the average time as:

 ⟨above z⟩=∫∞zdzmω(zm)τ(zm−z)/τ(zm)∫∞0dzmω(zm). (1)

The desired result is obtained for the distribution: :

 ⟨above z⟩=∫∞zdzmexp(−wzm/θ)τ(zm−z)∫∞0dzmexp(−wzm/θ)τ(zm)=exp(−wz/θ). (2)

In the above integrals going from to we have replaced by and introduced the variable , so that all the integrals go from zero to infinity and cancel out. Note that even though integral signs have been introduced no integration has been performed. We simply consider an integral as a sum of terms and employ the rule of addition associativity. We have employed also the fact that is the only function such that . Here is an energy introduced on the sole basis of dimension, later on proven to be a thermodynamic temperature. Thus a general form of the barometric law has been obtained without invoking the Boltzmann distribution. For a three-dimensional configuration corresponding to real atmospheres the same result holds: The -function is modified, but the barometric law does not depend on that function and therefore the result in (1) is unaffected. The more complicated case where depends on is treated at the end of the paper in Section 9. The result amounts to replacing in the above expression by the potential . This known result is obtained here most simply. The average forces exerted by the corpuscle on the cylinder lower and upper pistons are evaluated using a similar method in the following section 3.

## 3 Average force exerted by a corpuscle on pistons

We consider a unit-area cylinder with vertical -axis at some temperature. The bottom of the cylinder is located at altitude and a tight piston at altitude is free to move in the vertical direction. The cylinder contains a single corpuscle submitted to a force constant in space and time. For convenience we call the “weight” even though the force may not be of gravitational origin. The corpuscle motion may be relativistic or not. We evaluate the average force exerted by the corpuscle on the cylinder bottom and the average force exerted by the corpuscle on the piston. The following relation necessarily holds: . When the forces exerted on both ends are opposite: .

For the sake of comparison with textbook formulas note that in our one-dimensional model the average force corresponds to the pressure P, the height corresponds to the volume V, and . Then, in the non-relativistic approximation, the perfect-gas formula for a mole ( corpuscles): where , reads: where . Our result provides the ideal-gas law in a generalized form taking into account the weight . In that case, the force at the bottom of the cylinder exceeds in absolute value the force on the piston. We do not discuss the forces that the corpuscle would exert on the cylinder walls in a three-dimensional geometry.

### Perfect gases:

Since we have: . Let us recall the textbook result, and explain why it is not entirely satisfactory. The mechanical force exerted by the corpuscle on the piston is equal to the change of corpuscle momentum times the number of collisions per unit time , where the motion period: if the corpuscle speed is . Thus , where is some function of the energy : . We obtain the average force for a normalized energy distribution as: . To obtain a definite solution one usually makes the non-relativistic approximation and thus , where is the corpuscle mass, using the expression of the kinetic energy to which reduces in the present case. We obtain the textbook result: . The quantity on the right-hand side is identified with the temperature and we finally obtain: . This is the perfect gas law (often referred to as the “ideal gas law”) for a single corpuscle. However, the definition of given above is to a large degree arbitrary. For example, consider the ultra-relativistic case. Then where denotes the speed of light and . This forces us to redefine the temperature as being equal to instead of . In other words, the conventional formulation does not enable us to obtain the perfect gas law in a general form. Furthermore, the average value of depends on the distribution , which is presently unknown.

Alternatively one could call temperature the force that must be applied on the piston of a cylinder containing a gas such as helium to maintain at some fixed value. Then the empirical Boyle-Mariotte is established, and the Gay-Lussac law that asserts that is proportional to temperature follows from the definition of temperature just proposed. But if a real gas such as methane were employed instead of helium this empirical method of defining would lead us to conclude that is not a thermodynamic temperature. This is why, in the following, we suppose that . This seems at first to make the problem more complicated, but this is not so. At the end we may take the limit: and recover the usual perfect-gas law.

### Non-zero weight. Mechanical average:

The word “average” enters in this paper in two ways: as a mechanical average and as a thermal average. In the present paragraph we only consider mechanical averages. Physically, it is supposed that the cylinder ends have so much inertia that they do not respond to individual collisions. Let the corpuscle energy be denoted . The maximum altitude that the corpuscle would reach in the absence of the piston is given by: . The mechanical average force exerted by the corpuscle on the cylinder bottom is twice the corpuscle momentum when it collides with the plane, times the number of collisions per unit time , where denotes the motion period. When the corpuscle does not reach the piston, , we have . That is: . The corpuscle momentum is, according to the Hamilton equations (see Appendix A) given by . Thus: where is a constant. If we may set when , that is, at the top of the corpuscle trajectory. The corpuscle momentum at time is then , and the collision time is (see Figure 1). Thus . When the corpuscle possesses enough energy to reach the piston the motion period becomes: . Then the force experienced by the cylinder bottom is in general:

 {Fo(zm)=−wzm≤h,Fo(zm)=−wτ(zm)τ(zm)−τ(zm−h)zm>h. (3)

The force experienced by the piston when the corpuscle energy is , on the other hand, is likewise:

 {F(zm)=0zm≤h,F(zm)=wτ(zm−h)τ(zm)−τ(zm−h)zm>h. (4)

so that irrespectively of we have: . This means that the cylinder, considered as a rigid object of negligible weight, has an effective weight precisely equal to , a most intuitive result. By linearity, the same conclusion must hold for average forces: , for any energy distribution. This will be shown explicitly below for the appropriate energy distribution .

### Non-zero weight. Thermal average:

Because the cylinder lower end is in contact with a bath it suffers a slight thermal motion and the corpuscle energy slowly varies in the course of time. Accordingly, the average force experienced by the bottom of the cylinder and the average force experienced by the piston are respectively, from (3) and (4), if denotes the distribution:

 ⟨Fo⟩=∫∞0dzmω(zm)Fo(zm)∫∞0dzmω(zm)=−w∫h0dzmω(zm)+∫∞hdzmω(zm)τ(zm)τ(zm)−τ(zm−h)∫h0dzmω(zm)+∫∞hdzmω(zm). (5)
 ⟨F⟩=∫∞0dzmω(zm)F(zm)∫∞0dzmω(zm)=w∫∞hdzmω(zm)τ(zm−h)τ(zm)−τ(zm−h)∫h0dzmω(zm)+∫∞hdzmω(zm). (6)

According to our simplicity principle, the average forces must be independent of the corpuscle equation of motion, and thus of the -function. This condition obtains from (5) and (6) if one selects the following distribution:

 {ω(zm)=exp(−wzm/θ)τ(zm)zm≤h,ω(zm)=exp(−wzm/θ)(τ(zm)−τ(zm−h))zm>h, (7)

where has the dimension of an energy. The average forces become, using (5), (6) and (7):

 ⟨Fo⟩ =−w∫∞0dzmexp(−wzm/θ)τ(zm)∫h0dzmexp(−wzm/θ)τ(zm)+∫∞hdzmexp(−wzm/θ)(τ(zm)−τ(zm−h)) =wexp(−wh/θ)−1→{−θhwh≪θ−wwh≫θ ⟨F⟩ =w∫∞hdzmexp(−wzm/θ)τ(zm−h)∫h0dzmexp(−wzm/θ)τ(zm)+∫∞hdzmexp(−wzm/θ)(τ(zm)−τ(zm−h)) =wexp(wh/θ)−1→{θhwh≪θ0wh≫θ (8)

with: since: : The cylinder weight is irrespectively of the temperature (leaving aside the weight of the cylinder walls). Similar to what was done in Section 2, in the above integrals going from to we have replaced by and introduced the variable , so that all the integrals go from zero to infinity and cancel out. From now on we will omit the averaging signs on and .

For a collection of independent corpuscles having weights respectively, the force is a sum of terms of the form given in (3). In the case of zero weights (=0 or more precisely: ), the above expression gives: . Thus we have obtained the perfect-gas law: . The perfect-gas law does not depend on the nature of the corpuscles.

### Average force for a three-dimensional space:

We suppose that the cylinder radius is very large compared with and we do not consider the force exerted by the corpuscle on the cylinder wall. Motion of the corpuscle along directions perpendicular to (say, and ) does affect the round-trip time function . However, since the average force does not depend on this function, the ideal-gas law is unaffected. This is so for any physical system involving a single corpuscle provided the physical laws are invariant under a -translation (besides being static).

The internal energy, to be discussed in the following section, though, is incremented. One can prove that in the non-relativistic approximation and in the absence of gravity the internal energy is multiplied by 3. It would be incremented further by corpuscle rotation or vibration, not considered here. Using conventional methods, Landsberg [3] and Louis-Martinez [4] obtain exactly the same result as given above (except for the factor 3 in the expression of the internal energy, relating to the number of space dimensions considered).

## 4 Internal energy

The gas internal energy is the average value of , the gravitational energy being accounted for. Note that only corpuscule motion along the -axis is being considered. For simplicity, we first assume here that the cylinder rests on the ground level: . The expression of is, using the energy distribution given in (7):

 U =∫wh0dEEexp(−E/θ)τ(E/w)+∫∞whdEEexp(−E/θ)(τ(E/w)−τ(E/w−h))∫wh0dEexp(−E/θ)τ(E/w)+∫∞whdEexp(−E/θ)(τ(E/w)−τ(E/w−h)) =∫∞0dEEexp(−E/θ)τ(E/w)−∫∞whdE(E−wh+wh)exp(−E/θ)τ(E/w−h)∫∞0dEexp(−E/θ)τ(E/w)−∫∞whdEexp(−E/θ)τ(E/w−h) =(1−exp(−wh/θ))∫∞0dEEexp(−E/θ)τ(E/w)−wh∫∞whdEexp(−E/θ)τ(E/w−h)(1−exp(−wh/θ))∫∞0dEexp(−E/θ)τ(E/w) =(1−exp(−wh/θ))∫∞0dEEexp(−E/θ)τ(E/w)−whexp(−wh/θ)∫∞0dEexp(−E/θ)τ(E/w)(1−exp(−wh/θ))∫∞0dEexp(−E/θ)τ(E/w) =∫∞0dEEexp(−E/θ)τ(E/w)∫∞0dEexp(−E/θ)τ(E/w)−whexp(wh/θ)−1≡U1(θ)+U2(θ,h). (9)

If , one must add to the energy required to raise the cylinder from the ground level to , and .

Going back to the expression of in (4) we note that the first term minus corresponds to the kinetic energy , while the second term , plus , corresponds to the potential energy . In the non-relativistic limit, the first term, minus , gives the well-known expression , see the proof at the end of the present section. Without gravity we have of course . In the general case the splitting of into seems to be artificial. The internal energy thus is the sum of a term function of but not of and a term which tends to when . To evaluate the first term we need to know the round-trip time to within an arbitrary proportionality factor and an integration must be performed in that case.

The expressions given earlier for the average force in (3) and the internal energy in (4) may be written, setting , as:

 ⟨F⟩ =∂ln(Z)β∂hU=−∂ln(Z)∂β Z(β,h) =(1−exp(−βwh))w2πℏ∫∞0dzmexp(−βwzm)τ(zm). (10)

is the quantity called in statistical mechanics the partition function. The Planck constant introduced to make dimensionless plays no physical role in this paper. All the physical results may be derived from the above expression of .

### Non-relativistic approximation

In the special case of non-relativistic motion222Let us recall the following mathematical result:
, with , if is an integer, and . Thus: and: , so that: .
we have, see Appendix A: where denotes the corpuscle mass. Thus the first term in (4) is equal to and the expression of the internal energy reads:

 U(β,h)=θ(32−wh/θexp(wh/θ)−1). (11)

Let us give an example of application of the above formula. In a (Joule-Thomson) free expansion remains constant but the temperature may decrease. If the final cylinder height is infinite the new temperature is given by:

 ~θθ=1−23 wh/θexp(wh/θ)−1. (12)

The temperature may therefore be reduced up to the third of its initial value through unlimited free expansion. It is only for a perfect gas that the temperature remains constant.

When the non-relativistic approximation is made the expression of given in (4) becomes:

 Z(β,h) =(1−exp(−βwh))w2πℏ∫∞0dzmexp(−βwzm)2√2mzmw ≡(1−exp(−βwh))w2πℏ×f(β) =(1−exp(−βwh))w2πℏ2√2mw(βw)−3/2√π/2 =C(1−exp(−wh/θ))θ3/2, (13)

where is a constant ( and are constants) that will not be needed in the following because we will only be interested in derivatives of . For a perfect gas () to within an unimportant constant factor.

### Practical units:

The energy has been defined so far only to within a multiplicative factor from dimensional considerations. This factor is fixed by agreeing that exactly when the cylinder is in thermal equilibrium with water at its triple point. Here joules, is considered as an energy unit (akin to the calorie = 4.182… joules). This manner of defining is equivalent to the usual one, though expressed differently. The dimensionless quantity is the usual unit of thermodynamic temperature, expressed in kelvin. From our viewpoint it would be better to convene that exactly as the hydrogen triple-point temperature (HTP). The value of at the water triple-point (WTP) for example would be obtained experimentally by measuring the efficiency of a reversible heat engine operating with WTP as a hot bath and HTP as a cold bath. The known value is:

Next, measurements have shown that the number of atoms in 0.012 kg of carbon 12 is: . For this quantity of matter called a mole, the ideal-gas law therefore reads: , or: , with the ideal-gas constant: joules per kelvin per mole.

## 5 Stability:

Solutions obtained for the force and the energy imply stable equilibria provided two conditions be satisfied. Firstly, the isothermal compressibility must be positive. This is readily verified since the derivative of the force given in (3) with respect to is negative. Secondly, one must verify that the isochore heat capacity is positive. This is a more difficult problem solved below. Given that and are positive it follows that the isobaric heat capacity is positive, and the isentropic compressibility is positive also. Thus, let us show that is positive.

### U is non-negative:

Assuming that is derivable and an integration by parts of the numerator of in (4) gives

 ∫∞0dEexp(−βE)τ(E/w)=ϕ(β)β,ϕ(β)≡1w∫∞0dEexp(−βE)τ′(E/w). (14)

Since is non decreasing, is non-negative and is negative. Thus

 U1(θ)=−ddβlnϕ(β)β=θ−ϕ′(β)ϕ(β)>θ. (15)

Since , we have proven that: is positive.

### C is non-negative:

We have

 ∂U2(θ,h)∂θ =−(α/2sinh(α/2))2>−1,α≡whθ, U′1(θ) =1+β2d2dβ2lnϕ(β). (16)

In order to get , it thus suffices to show that , that is, letting for short,

 (∫∞0dEE2f(β,E))(∫∞0dEf(β,E))≥(∫∞0dEEf(β,E))2. (17)

Since is non-negative (17) is the classical inequality regarding the moments of order 0, 1 and 2 of the mesure defined by: . Thus: and the expressions obtained from our simplicity principle imply stability of the equilibria.

## 6 The Helmholtz fundamental relation.

It is convenient to introduce the Helmholtz fundamental relation: . The letter originates from the German “Arbeit” or work, but this letter may also stand for (constant temperature) “Available work”. The force that the corpuscle exerts on the base, the force that the corpuscle exerts on the piston and the internal energy result from the Helmhotz fundamental relation depending separately on and . We consider thus a cylinder whose base has been raised from to . The previous relations for in (3) and for in (4) may be written as:

 Fo =wexp(−wh/θ)−1=−∂A∂ho,F=wexp(wh/θ)−1=−∂A∂h1 U =A−θ∂A∂θ=∂(βA)∂β A(θ,h0,h1) =−θ(ln(1−exp(−wh/θ))+ln(w2πℏ∫∞0dzmexp(−wzm/θ)τ(zm))+who (18)

with . Thus, if the cylinder bottom is raised to an altitude , and are both incremented by . From now on we set for simplicity unless specified otherwise.

We have obtained an expression for the Helmholtz fundamental relation for the special case of a single corpuscle submitted to a constant force in the canonical ensemble. This fundamental relation has the same mathematical and physical content as the often-used energy fundamental relation: and the entropy fundamental relation: , see [8]. The following expressions therefore coincide with the conventional ones applicable to any working substance.

### The energy θ is a thermodynamic temperature

We prove in this section that , introduced in previous sections on dimensional grounds only, is a thermodynamic temperature. We do this by showing that the efficiency of a reversible thermal cycle employing ideal gases is: , where is the cold-bath temperature and the hot bath temperature: this is the accepted Kelvin definition of absolute temperatures.

From the law of conservation of energy the heat released by the gas is from (6):

 −δQ≡dU+⟨F⟩dh=dA−∂A∂θdθ−∂A∂hdh−θd(∂A∂θ)≡θdS,S=−∂A∂θ. (19)

For any function such as : . We employ only two independent variables namely and so-that partial derivatives are un-ambigous. If the gas is in contact with a thermal bath (=constant) is the heat gained by the bath. The quantity defined above is called “entropy”. In particular, if heat cannot go through the gas container wall (adiabatic transformation) we have that is, according to the above result: . Thus reversible adiabatic transformations are isentropic. Note that , here defined as the ratio of two energies, is dimensionless. It may therefore be written as the logarithm of a dimensionless quantity. The fact that defined above is a state function suffices to prove that is a thermodynamic temperature as shown below.

### The Carnot cycle:

A Carnot cycle consists of two isothermal transformations at temperatures and , and two intermediate reversible adiabatic transformations (). After a complete cycle the entropy recovers its original value and therefore . According to (19): , and therefore . Energy conservation gives the work performed over a cycle from: . The cycle efficiency is defined as the ratio of and the heating supplied by the hot bath. We have therefore: , from which we conclude that is the “thermodynamic temperature”. Since Kelvin time, thermodynamics temperatures are strictly defined from Carnot (or other reversible) cycles efficiency. In practice, temperatures may be measured by other means and employed in other circumstances.

We have implicitly assumed in the above discussion that the working medium (presently an ideal gas) has reached the bath temperature before being contacted with it. Otherwise, there would be at that time a jump in entropy, and the cycle would no longer be reversible. Given initial values, the temperature change for an increment in the isentropic regime () follows from the relation: , where may be expressed in terms of , (4), from the above expressions. The details will be omitted. It suffices to know that may be varied by varying , in a calculable manner, in an isentropic transformation.

## 7 Expressions of the entropy

In (19) we have expressed the state function in terms of the Helmholtz potential

 S=−∂A∂θ=∂(θln(Z))∂θ=βU+ln(Z(β,h)) (20)

using for the expression given in (6).

### Perfect gases:

For a perfect gas (, non-relativistic approximation) the expression of in (4) may be written as:

 Z(β,h) F =−∂A∂h=−1β∂(βA)∂h=θhU=∂(βA)∂β=θ2 (21)

These are the usual expressions for the equation of state and the internal energy of a perfect gas.

According to (20) and (7) the entropy is:

 S=−∂A∂θ=∂(θln(hθ1/2))∂θ=ln(hθ1/2)+12. (22)

Since , the fundamental entropy relation reads:

 S(U,h)=ln(h√2U)+12. (23)

We recover from this expression again: , and: , that is .

### Ideal gases:

We consider a non-zero corpuscle weight . The procedure is the same as in the previous paragraph. Let us collect previous results in (4) and (11) for the internal energy , in (20) for and (4) for the entropy , applicable to a cylinder of height :

 U S =U/θ+ln(Z)≈(32−wh/θexp(wh/θ)−1)+ln(1−exp(−wh/θ))+32ln(θ) (24)

in the non-relativistic approximation, to within a constant that makes the quantities dimensionless if desired. The fact that the entropy tends to - when should not be a cause for concern because only entropy differences are considered.

### Another form of the entropy:

Recall from (20) that:

 S≡−∂A(β,h)∂θ=−β∂ln(Z(β,h))∂β+ln(Z(β,h)). (25)

The successive trajectory actions are discretized with spacings equal to the Planck constant, and thus the corresponding energies are correspondingly discretized with subscripts and we suppose that different values correspond to different energies. In the present classical paper the Planck constant is allowed at the end to assume arbitrarily small values.

Let the function be written as a sum of terms instead of an integral. Then the entropy may be written as:

 S=−∞∑k=0pk(β,h)ln(pk(β,h)),pk(β,h)=exp(−βεk(h))Z(β,h),Z(β,h)=∞∑k=0exp(−βεk(h)), (26)

as one readily verifies by substituting the expression of into the expression of . The above is a simple mathematical transformation. However, when there is some uncertainty concerning the presence of a corpuscle in the cylinder, it is useful to interpret the as independent probabilities. The entropy may not decrease when constraints are removed or restored inside a thermally isolated body[9]. Let us quote these authors: “A common formulation of the law of increase of entropy states that in a process taking place in a completely isolated system the entropy of the final equilibrium state cannot be smaller than that of the initial equilibrium state. This statement does not specify that thermal isolation is all that is needed for its validity, with no need for mechanical isolation”.

Let us consider now two boxes labeled “A” and “B” and a single corpuscle. If the corpuscle is in box “A” the probability that the level be occupied is denoted with: . If the corpuscle is in box “B” the probability that the level be occupied is denoted with . When the corpuscle is in box A with independent probability and in box B with probability , the should be multiplied by and the should be multiplied by . We therefore have for the entropy in that case:

 ~S =−∞∑k=0pkAPAln(pkBPA)−∞∑k=0pkBPBln(pkBPB)=SAPA+SBPB+ΔS SA =−∞∑k=0pkAln(pkA),SB=−∞∑k=0pkBln(pkB),ΔS=PAln(1/PA)+PBln(1/PB). (27)

The additional term accounts for the fact that it is not known with certainty whether the corpuscle is in box A or in box B. When the two boxes are identical () we have: .

## 8 Change in entropy upon removal and restoration of a separation

The process presently discussed is often considered in relation with the so-called “ Gibbs’s paradox”, see for example[10]. Let us give an adapted quotation of his paper: “A new state is created by inserting a partition into the 2V volume. It is clear that there should be no change in entropy in that process. The initial state was prepared knowing precisely which distinguishable particles were in each subvolume, but there is no such knowledge about the final state, so from the information point of view it is clear that the initial state and the final state are not equivalent”. We consider a thermally-isolated vertical cylinder of height (from to ) separated by an impermeable wall at altitude . The lower part of the cylinder is labeled “A” and the upper part is labeled “B”. A corpuscle is introduced in part A. The separation is removed: The internal energy is unchanged since the system is adiabatic and no work has been performed. Because is an increasing function of temperature and a decreasing function of the cylinder height, doubling the height amounts to reducing the temperature to a value that we denote . Then we restore the separation and show on the basis of our formulas (derived from the simplicity principle) that the total internal energy remains the same and that the total entropy also remains the same, in agreement with the first and second laws of thermodynamics. We treat first the simple case of perfect gases (no weight, non-relativistic approximation) and then the case of ideal gases (non-zero weight , arbitrary hamiltonian function).

### Perfect gases:

For a perfect gas the entropy is, omitting unimportant constants: since part B does not contain any corpuscle, with the internal energy and the constant-volume heat capacity. Next we remove the separation. This can be done without any work employed and thus without any change of . The new expression of the entropy is therefore: since is unchanged:

 S2=S1+ln(2). (28)

Let us now suppose that the impermeable separation is restored. Again, this can be done without changing the internal energy and thus the temperature. If we knew that the corpuscle is in part “A”, the entropy would be reduced to . However, there is a probability that the corpuscle be in part “A” and a probability that the corpuscle be in part “B”. Thus, according to (7) the entropy is now:

 S3=S1−PAln(PA)−PBln(PB)=S1−12ln(12)−12ln(12)=S1+ln(2)=S2. (29)

Thus the entropy has not been reduced by restoring the separation. In the present model the entropy remains the same. Since we are dealing with a single corpuscle the problem of corpuscle distinguishability does not arise.

### Ideal gases:

Let us recall the general expressions given earlier for the internal energy in (11) and for the entropy in (4) for a cylinder of height at a temperature reciprocal :

 βoU(βo,ho)=βoU1(βo)−αoexp(αo)−1≡e(βo)−αoexp(αo)−1,αo≡βowho S(βo,ho)=βoU(βo,ho)+ln(Z(βo,ho)) =(βoU1(βo)−αoexp(αo)−1)+ln(1−exp(−αo))+ln(f(βo)) ≡−αoexp(αo)−1+ln(1−exp(−αo))+g(βo) (30)

for some functions of , to within constants that makes the quantities dimensionless if desired. The fact that the entropy tends to - when should not be a cause for concern because only entropy differences are considered. We have omited the term in the expression of .

In the process of removing the separation the internal energy remains the same but the cylinder height becomes , and thus the temperature reciprocal assumes a new value that we denote . The parameter in the above expression of the internal energy becomes , the value of being given by the relation:

 βU=e(β