The following work remained so far an Institute Report; it is KFKI-1994-14. I did not get any comment from anybody; and I had other things to do. But I think it deals with a fundamental question, so I broadcast it.

Briefly: if you believe that Callen’s Axioms are full, you had better read either K. Martinás’ seminal (sorry for the pun which you may not understand) paper which is Ref. 6 below, or this HTML. When I speak about "completeness", I do not mean something shown by Gödel. I mean that Callen’s Axioms do not fix fully the thermodynamic description. And Laws from 0^{th} to 3^{rd} do not make it fully defined either.

We showed this in Ref. 1 in 1986. Dr. Martinás manufactured a fifth Axiom as a 4^{th} Law already in 1980, and it narrowed very much the ambiguities, although not completely. So in 1984 Dr. Martinás & myself manufactured a sixth Axiom as 5^{th} Law; that is Ref. 3 below except that it was originally a Report, KFKI-1984-33.

But still some fredom remained. That is discussed here. Discussed, but not fully solved; as you will see, there are still problems.

I admit thast this work did not trigger (so far) any comment at all, not even from K. Martinás. And then?

The HTML file has been made from the original Word file, with trivial changes advisable for HTML files.

KFKI-1994-14

ON THE ZERO POINT OF THE CHEMICAL POTENTIAL

B. Lukács

Central Research Institute for Physics, H-1525 Bp. 114. Pf. 49., Budapest, Hungary

ABSTRACT

It is sometimes ignored but true that zero points of the entropic thermodynamic intensives cannot be fixed by thermodynamic measurements. Constant universal yero-point shifts leave all equilibrium configurations unchanged. Therefore the zero points are matters of thermodynamic *conventions*, and calculations can be performed in any of the conventions.

However it turned out earlier that simple thermodynamic *principles* can be formulated to fix the zero points of the intensives 1/T and p/T (except for exceptional systems). Other conventions are still possible, but the preferred ones are "simpler" from thermodynamic viewpoints. Now we discuss the problem for the further intensives -µI/T, and give the appropriate principles at least for "elementary" particles.

1. INTRODUCTION

Thermodynamics is a phenomenologic way to describe the macroscopic behaviour of matter, at least in the absence of long-range self-interactions. It is very economic in some cases even under such exotic situations as highly compressed nuclear matter or Grand Unification era of the early Universe.

Thermodynamics has a nontrivial relationship with statistical physics. The latter one is often regarded as the microscopy behind the former. However great caution is needed about such a statement. The statistical physics of most systems familiar in thermodynamics is unknown or at least not complete. The simplest example is the van der Waals system. It is the textbook example of a thermodynamic system with two phases; in the same time its momentum distribution function is unknown, although the *first order* deviations from the ideal gas pressure can be derived from a reasonable two-particle distribution. One can rather visualize the relationship as two domains for the problems tractable for the two particular disciplines, overlapping in a small region of ideal gases, Ising-type systems and such (true, in the overlap domain statistical physics is indeed the microscopy for the other).

Therefore for complicated systems it is worthwhile to build up thermodynamics on its own basis. The full thermodynamic description of a system of well defined set of the independent extensive variables means the determination of its entropy function, but, perhaps surprisingly, the task cannot be fully executed by thermodynamic measurements. Some universal freedom is left, which does not affect any thermodynamic statement [1], but does the mechanical, gravitational, &c. behaviour of the system [2]. Therefore, different thermodynamic *conventions* are possible. If this diversity is disturbing, one has an alternative. Either that convention is to be selected which is compatible with the other disciplines (e.g. the thermodynamic pressure be as similar as possible to the mechanical one), or the freedom is to be removed by thermodynamic *principles* which are simple, transparent and conform with common sense (zero point fixing). In an earlier paper [3] this was partially done in connection with the pressure of the systems. However there are free parameters not influencing the pressure, those directly connected to the zero points of the (entropic) chemical potentials.

The zero point of a chemical potential is always a matter of convention in some sense. Namely, energy zero points can be shifted in some correlated way. In addition, conventions based on ideal distributions (either Boltzmann or Bose or Fermi ones) differ between nonrelativistic and relativistic treatments due to rest energy even if the energy zero point is unchanged. Such a difference cannot affect the dynamics of the system, but it does influence particle transmutation, creation and annihilation equilibria and rates. While sound physical intuition generally eliminates the ambiguity, under exotic circumstances some confusion is not impossible and does occur. Therefore it is not unnecessary to discuss the question of *thermodynamic* zero points of chemical potential-like quantities.

This paper is devoted to the thermodynamic freedom of the entropy function connected to the chemical potentials. It is a direct continuation of Ref. 3. As it will turn out, the question is too complicated and the known facts do not seem sufficiently general and restricting to completely finish with the problem. Still, for the case of conserved elementary particle components a convention will be proposed which completely fix the free parameters and is conform with common sense.

Sect. 2 states the problem in general. Sect. 3 recapitulates earlier results for the zero points of 1/T and p/T. Sect. 4 briefly lists the main properties of possible particle degrees of freedom. In Sect. 5 the form of the zero point fixing for -µ/T is formulated. Sect. 6 gives the available constraints on the constants between conventions preferred by thermodynamic and hydrodynamic simplicity. Sect. 7 outlines the problem for two, interacting, particle components and Sect. 8 does the same for a composite particle. Sect. 9 gives a bird's beye overview of the particle components in the early Universe, while Sect. 10 tries to summarize the conclusions.

2. THE PROBLEM STATED

As told above, we want to build up the thermodynamic description of systems by pure thermodynamic tools. It goes step by step. First the set of the independent macroscopic variables is to be selected. This procedure has been defined by Landsberg [4]; it contains argumentation on macroscopical distinguishability of similar systems and will not be repeated here. For simplicity here we *assume* that the complete set of independent macroscopic parameters contains the following *extensives*:

{X^{i}} = (V,E,N^{I})

(2.1) I = 1,...,M

i = ø,0,I

that is volume, internal energy and some particle numbers. This is not always true, but it is so general that sometimes is put as an axiom [5]. Furthermore we assume that there is no problem in defining and measuring the extensives (which is not quite trivial).

The total thermodynamic information is contained by the *entropy* function S(X^{i}) [5], which is of homogeneous first order:

(2.2) X^{r}¶S/¶X^{r} = S

where, and throughout the whole paper, we use the Einstein summation convention (automatic summation for indices occurring twice, above and below). Indeed, the derivatives

(2.3) Y_{i} º ¶S/¶X^{i}

are homogeneous zeroth order functions of the extensives, and will have homogeneous spatial distributions in equilibrium [5], so Y_{i} are the *intensives*.

Now we should determine the entropy function S(X^{i}) of the system under investigation. In thermodynamics S cannot be measured. (In statistical physics it would be directly connected to the number of microstates.) However all its first derivatives Y_{i} influence the thermodynamic equilibrium. Therefore one may try with establishing equilibrium states and measuring the extensives X^{i} there. For this a set of different material systems are needed. Let us take a set a=1,..., of different material systems which have the same kinds of particle numbers as independent variables. Then, establishing infinite numbers of equilibria one gets equations of form

Y_{ai}(X_{a}^{k}) = Y_{ßi}(X_{ß}^{k})

(2.4) Y_{ai}(X^{k}) º ¶S_{a}(X^{k})/¶X^{i}

a = 1,...,q

and then a set of differential equations is obtained for the S_{}a functions. The system consists of ½q(q-1)(M+2) equations, so may become overdetermined.

As it has turned out [6], in some sense the extractable thermodynamic information does not increase for q>3; there the overdetermination can be used only for check and reducing observational errors. If at least 3 systems of different entropy functions are brought into equilibria at infinitely many different intensive values, then the entropy functions of all the systems can be calculated up to free constants:

(2.5) S_{a}(X^{i}) = K²{S°_{a}(X^{i}) + AV + BE + C_{R}N^{R}}

where {S°_{a}(X^{i})} is a particular solution and the parameters K, A, B, C_{I} are constants. These constants are the same for each system, so they belong to the particular thermodynamic convention. As told above, this freedom remains unchanged for q->¥.

From the result it is easy to understand the freedom. The parameters cause constant shifts and/or scale changes in the entropic intensives, independently of a. Therefore all the equations Y_{ai}=Y_{ßi} transform into themselves.

The existence of such a freedom might be regarded as a shortcoming of the thermodynamic approach. However, the same freedom would appear in statistical physics as well (see the Appendix). We note that a Riemannian structure exists in the thermodynamic state space [7], [8], [9], which, if defined via the entropy S, possesses direct statistical and indirect thermodynamical meanings [9], [10], [11]. Namely, using the extensive *densities* x^{i}ºX^{i}/V as coordinates, a metric is defined by

(2.6) g_{ik} = -¶²(S/V)/¶x^{i}¶x^{k}

and then in any other coordinate system the metric is given via the invariant line element:

(2.7) ds²=g_{rs}dx^{r}dx^{s}=g_{rs}'dx^{r}'dx^{s}'=ds'²

Now, Einstein's fluctuation formula gets for small fluctuations the simple form

(2.8) <dx^{i}dx^{k}> = V*^{-1}g^{ik}

where V* is the volume of the subsystem monitored. It turns out that the Riemannian geometry yields fluctuation probabilities even for smaller subsystems where (2.8) is the leading term. One can get the probability distributions fully from g_{ik}(x^{l}); therefore the entropy function S(X^{i}) determines not only the macroscopic thermodynamics but the fluctuations as well. Via the fluctuations one can get paths of minimal length between thermodynamic states, can define natural regions in multiphase systems and so on.

Then, the transformation (2.5) leads to

s_{a}(x^{i}) = K²s_{a}o+A+Be+C_{R}n^{R}

s = S/V

(2.9) e = E/V

n^{I}= N^{I}/V

and, therefore,

(2.10) g_{ik} -> K²g_{ik}^{o}

Now, consider two metric tensors:

(2.11) g_{ik}(x), K²g_{ik}(x)

These two Riemannian structures are indistinguishable. Namely,

(2.12) ds² -> K²ds²

and now ds² has the dimension 1/(volume). Therefore the value of ds² is invariant if simultaneously with the K² transformation the volume unit is rescaled too. Since there is no "natural" volume unit, this is always possible. Instead of further argumentation we simply refer the theory of Riemannian spaces, in which two metric tensors describe the same space if they differ only in a constant multiplicative factor [12].

So the transformation (2.5) does not affect either macroscopic thermodynamics or fluctuations; if one description is correct, the others are correct as well. The freedom exists, and it is indeed the freedom of defining the zero points of the entropic intensives.

Consequently *thermodynamics* can work without further discussion of the constants. However the appearance of free constants is rather inconvenient. A great caution is needed not to use different conventions in different measurements. In addition thermodynamics contains a quantity called *pressure* and a similar quantity appears e.g. in hydrodynamics. There is no a priori reason for the equality of these two pressures, and, indeed, in the presence of e.g. viscosity the two quantities have different variables, so they cannot be equal. However one would expect as high similarity as possible, i.e. that they agree in "leading terms". But the thermodynamic pressure depends on the free constants, so the degree of similarity depends on the (arbitrary) thermodynamic convention.

Then even to be able to speak clearly, one must define the actual thermodynamic conventions. **C**° is the one where S° of eq. (2.5) was calculated; let it be a convention selected by some thermodynamical principles (which will be discussed later). An alternative one will be **C**' which is "nearest to" other physical disciplines, e.g. where the thermodynamic pressure is as close to the hydrodynamic or cosmologic one as possible. In addition, various subjective conventions **C**^{1}, **C**2,... may exist according to everybody's own preference. If **C**° is defined by any unambigous, reasonable way according to common sense, then afterwards unambigous statements can be done either so that we make all the statements in **C**°, or so that we work in a definite other **C**^{i}, but give the specific constants between the two conventions as well. Any statement in one convention can be transformed into any other one, if **C**^{i} and **C**^{k} are precisely defined. In what follows we try to make a reasonable proposition for **C**° in a language conform with axiomatic thermodynamics, and when it is done we compare **C**° and **C**', the latter singled out by the whole Universe.

For definiteness, we give the transformational rules from (2.5) for the particular intensives. According to the scheme (2.1) the *entropic* intensives are named as

{Y_{i}} = (p/T, 1/T, -µ_{I}/T)

(2.13) i = ø,0,I

where p is the pressure, T is the temperature and µ_{I}'s are the chemical potentials. Then going to a general **C** eq. (2.5) leads to

T = K^{-2}T°/(1+BT°)

(2.14) p = (p°+AT°)/(1+BT°)

µ_{I} = (µ_{I}°-C_{I}T°)/(1+BT°)

where the upper ° denotes the intensives of **C**°.

Hence one can see that K trivially sets the scale unit for the temperature. Henceforth we ignore that constant. The (thermodynamic) pressure depends only on A and B. Therefore the constants C_{I} do not have any dynamical consequence, and this is the reason that the study of Ref. 3 did not extend to them.

3. ZERO POINTS FOR 1/T AND p/T

In this Chapter, for analogy, we recapitulate earlier zero point conventions for T and p.

Temperature

Based on an idea of Guggenheim [13] Martinás proposed that

(3.1) lim_{E->¥} (¶S/¶E) = 0

be required [6]. According to eqs (2.5) and (2.7) this fixes the constant B if required *for a definite system*. So the selection of that particular system *plus* Cond. (3.1) fixes the B of the particular convention. The meaning of Cond. (3.1) is infinite temperature at infinite energy density, which is reasonable enough, although, as we have seen, not obligatory. If the Callen postulates [5] held, then Cond. (3.1) could be required for *all* the thermodynamical systems. Namely S generally increases slower than linear with E, and this is one of the Callen axioms; then at E->¥ the leading term is universally the "artificial" BE, and then (3.1) puts the universal B to 0 in **C**°. (*Note*: henceforth a limes ->¥ is meant in such a way that all the other extensives remain finite.)

However, *exceptional* model systems are known. E.g. if a system can take only finite internal energy, then the limit in (3.1) does not exist [3]. In all known examples such systems have T<0 states at high energies. The simplest example is the case of fixed magnetic dipoles; a good physical realization is the magnetic excitation of LiF [14].

Such *exceptional systems* may exist. However in axiomatic thermodynamics one must unambigously define "exceptions". Luckily the known exceptional systems do not obey the Callen axioms [5], and in the same time the limit (3.1) does not exist for them, since in them E has a finite upper bound. So Cond. (3.1) can define the B of **C**° in spite of the *known* exceptional systems.

The only problematic system would be such a one, which possesses a term linear in E in the convention when other systems do not. Then this system would have a limiting asymptotic temperature at infinite energy, while the others would not. Then it would not fulfil the Callen axioms. If such an exceptional system exists, then it must definitely be excluded when Cond. (3.1) is required. Note that in that asymptotic domain thermodynamic stability ceases (the specific heat matrix degenerates).

The existence of systems with finite maximal temperature were sometimes discussed, see e.g. Hagedorn's idea for the "melting point of the vacuum". Such a system would prevent the further heating of all the systems in thermal connection with it. If all the systems were to possess a universal limiting temperature, then we are still not in **C**°, and the universal limiting temperature can be transformed into ¥ by (2.5). If some systems were to possess limiting temperatures but not universally, then all having such can be called special systems, and we can turn to the others when requiring Cond. (3.1).

Pressure

Per analogiam, for fixing A one can require

(3.2) limV_{->¥} (¶S/¶V) = 0

as was done in Ref. 3. Its meaning is: pressure vanish in infinite dilution. Again for a particular system (3.2) would fix A. As for the generality, one may observe that familiar systems have a common tendency for p->0 when diluted infinitely. For all these systems the leading term of (2.5) in V is AV in a specific convention, and then Cond. (3.2) fixes A.

Quite physical systems are, however, known whose vacuum energy density is a *positive* constant e_{o}, best examples are (perturbative) Quantum Chromodynamics and the symmetric phase of Grand Unification. If so, then at infinite dilution p->-e_{o} [15]. However simultaneously E->e_{o}V->¥, i.e. the limit needed in (3.2) is impossible. So *these* systems do not disturb the operativity of Cond. (3.2), because for them eq. (3.2) cannot fix anything.

Cond. (3.2), however, would fix A differently for different systems if such systems existed for which e_{o}<0. Then p->-e_{o}>0, the leading term in (2.5) would be (A-e_{o})V, and Cond. (3.2) would fix A in an e_{o}-dependent way.

No such physical or model system is known up to now. However again no physical law is known prohibiting this behaviour. Still, such systems could easily be recognized as exceptional ones. Namely, they would infinitely expand into vacuum or into any other low pressure systems, because of the limiting *positive* pressure. The boundary surface would propagate with light velocity in interstellar space, a phenomenon never seen by astronomers. (One might contemplate about a "survival of fittest", namely that our vacuum has already expanded into the others, because its eo is the lowest. If this were the reason, no e_{o}<0 system would turn up.) Again, then, we can leave out *exceptional* systems when applying Cond. (3.2).

Now we turn to particle numbers and chemical potentials. However in general this goal would exceed the frameworks of one paper, since the systems considered may possess any particular number of particle components, these components can be disjoint or can transmute into each other and they may or may not be conserved. In the next Chapter we briefly discuss these possibilities, although mainly this paper will be devoted to a *single* *conserved* component.

4. ON PARTICLE DEGREES OF FREEDOM

Particle numbers, as extensives, differ in rôle from volume or internal energy. Let us compare V and N.

There is always one volume for a thermodynamic system, while it may have more than one independent particle components. If there are more than one of them, they may or may not be transformable into each other. Finally, the volume obeys a kind of "balance law": a hydrodynamic term is making V to change. A particle number may also obey a balance law with or without source term ("conservation); however another possibility is that the other extensives prescribe the momentary value of the particle number.

In this paper we want to concentrate on a *single* component but will note some complications for the more general case. In the most general case transmutations are possible. Now, consider for example the ground state and first excitation of H atoms. Then their equilibrium ratio is determined by the remaining extensives, so in equilibrium the second particle number is not an independent quantity, and here we do not want to deal with nonequilibrium thermodynamics. And this argumentation holds as well for the alternative of balance law or not: if a particle number is determined by the other data, then it is not an independent quantity.

Therefore now we turn to a single particle component N with a balance law; and when a possibility of generalization will be mentioned, it will be two components, N^{1} and N^{2}, with independent balance laws without transmutation.

5. ON THE ZERO POINT OF µ/T

We would like to copy the constructions (3.1-2). However absolutely the same structure would not do. Namely, particle-particle interactions, or simply quantum statistics lead to

(5.1) lim_{N->¥} µ = ¥

therefore at N->¥ the leading term of (2.5) in N is *not* CN. Consequently at N->¥ a prescription for the first derivative of S does not fix C properly.

We, however, propose a similar condition with a minimal alteration:

(5.2) lim_{N->0} (¶S/¶N) = (something)

At first sight this condition seems to be equally impossible, but in this Chapter we show that a condition of type (5.2) is meaningful. Anyway, the "particle-free" limit of a system must be something simple. For phenomenologic "particles" such that water droplets, billiard balls etc. no general strong support is given to a "natural" zero point, even in the particle-free limit, by general principles. However such entities are composed of "more elementary" particles. So let us use first "elementary" ones.

For a nonrelativistic Boltzmann gas lim_{N->0} µ = -¥, and it is so for an ideal Fermi gas as well. That seems sufficient; interactions may be negligible at N/V=0. If µ->±¥, then one cannot fix C in (2.5) by means of a condition of structure (5.2). And the infinite limiting value remains unchanged for the systems usually called *relativistic* ideal gases. However this behaviour is unphysical, as will be shown.

Notice that by switching on relativity a new phenomenon called pair creation appears. There are two cases: any particle either has an antiparticle, whose quantum numbers are just opposite, or it is its own antiparticle as well. For the first case we may have the electron e- and the positron e+ as example; for the second case the simplest example is the photon g.

With characteristic quantum numbers the particle will obey a balance law; e.g. electric charge conservation prevents electrons to be created at will. However the particle-antiparticle pair has the same charges as the vacuum, thus the process

(5.3) e^{+}+e^{-} <--> ø

is permitted. But if so, then *in* *equilibrium* both numbers of particle and antiparticle are not independent, since no conservation constraint may exist for the pairs. As it turns out,

{N° of ptcle - N° of antiptcle} is the extensive N

{N° of ptcle + N° of antiptcle} is prescribed by other extensives

Indeed, for a relativistic Fermi gas with or without interaction one gets Fermi distributions with corrected energy, with µ for the particle and with -µ for the antiparticle [16].

But then the usual "ideal relativistic Fermi gas" without antiparticles is *not* an equilibrium distribution and does not belong to equilibrium thermodynamics. Nonequilibrium situations may be very interesting, and even relevant for timescales short compared to characteristic times of reactions; however the problem investigated in this paper is connected to equilibria, and we want to remain restricted to that. For a particle which is its own antiparticle the analogon of Reaction (5.3) reads as

(5.4) ng <--> mg

where m and n are arbitrary nonnegative integers. Hence in equilibrium the actual particle "number" is determined by the independent extensives and it does not constitute an independent extensive parameter.

Consequently here we may restrict ourselves to particles *with* antiparticles, and then the proper extensive N is the *net* number. Now, observe that in the "naïve" convention ideal relativistic Fermi gases lead to

N = 0 at µ = 0.

This suggests the actual form

(5.5) lim_{N->0} (¶S/¶N) = 0

for the condition (5.2). If S's linear term in N were universal at N=0 then Cond. (5.5) would fix the constant C in (2.5) and then, apart from the trivial factor K, the thermodynamic convention **C**° would be fixed by Conds. (3.1), (3.2) and (5.5). However, first this universality should be proven.

Remember that in this Chapter we deal with a *single* particle number. By other words, the present study concentrates on systems having the *same* set of extensives {V,E,N}. This means that for all systems considered the only particle component is the *same* kind of particle, which, in addition, is "elementary". In this context looking for universality means that we investigate different systems composed of the same kind of particles. The linear term may be universal for one kind of particles and may not for another.

For simpler formulation let us fix the actual particle to a symbolic R, anything may it be, with an antiparticle. Then

(5.6) N º N(R) - N(R_{anti})

Then we can prove some Theorems about the universality of Cond. (5.5). (Note that we cannot *prove* (5.5) itself: a finite constant on the right hand side is a matter of convention. However for families of systems we can see if they are transformable simultaneously to (5.5).) These Theorems will not be proven with mathematic rigour; however I think if needs be, any Proof could be worked out in the necessary details.

*Theorem 1: *Cond. (5.5) can be simultaneously fulfilled for all systems composed only of R if this particle is a subject of C symmetry.

*Proof:* With C symmetry configurations must be of equal probability with antiparticles instead of particles and vice versa. Then S(V,E,N)=S(V,E,-N) with an extremum, therefore, at N=0.

*Exceptional situations:* The above argumentation may be invalid in the presence of an external field preferring R to R_{anti} or inversely. In this case, however, the system under investigation is not closed and volume forces are present, which often may lead to problems in thermodynamics.

*Comment:* C symmetry is naïvely expected for systems, but is violated in weak interactions.

If Theorem 1 cannot help, we can turn to a

*Theorem 2:* Cond. (5.5) can be simultaneously fulfilled for all macroscopic near-equilibrium systems composed only of R if this particle is a subject of CP symmetry, unless some spatial spontaneous symmetry breaking is present.

*Proof:* In a macroscopic system without built-in anisotropy or long-range structure the parity is averaged out, because it belongs to the spatial configuration. So P is a symmetry by averaging, therefore microscopic CP symmetry implies macroscopic C symmetry for the system, and then we are back at Theorem 1. For a subsystem this argumentation could be invalidated by the anisotropies via gradients in the neighbourhood. However, first order small gradients cause second order deviations in the thermodynamic potentials. Therefore in the limit of "local equilibrium" or "cellular equilibrium" the relevant quantities do not exhibit the effects of gradients; then again the macroscopic averaging results in a C symmetry.

*Exceptional situations:* The above argumentation may be invalid in the presence of an external field preferring a direction (see gravity, electromagnetic forces, &c.). In this case, however, the system under investigation is not closed and volume forces are present, which often may lead to problems in thermodynamics. Macroscopic quantum coherence may occur in special systems (ferromagnets, superconductors, &c.); they need individual attention and may be exceptional.

*Comment:* CP symmetry is violated in *hyper*weak interaction. The only known example up to now is the kaon. So the present argumentation may be invalid for a system of kaons (and antikaons). If this CP violation is a consequence of external hyperweak fields [17], then we are back at the Comment of Theorem 1. If the CP violation comes from a *spontaneous* symmetry breaking [18], then the symmetric configuration still may be an extremum although not ground state. Anyway, the hyperweak CP violation deserves further attention, but this will not be made in this paper.

If Theorem 2 cannot help, we can turn to a

*Theorem 3:* Cond. (5.5) can be simultaneously fulfilled for all macroscopic systems in equilibrium, composed only of R if this particle is a subject of CPT symmetry and if spatial spontaneous symmetry breaking is not present.

*Proof:* In equilibrium a macroscopic system shows symmetry for T reflection, so then microscopic CPT symmetry implies macroscopic CP symmetry for the system, and then we are back at Theorem 2. For evolution through equilibrium states we can use the familiar quasistationary limit: if the change rate is moderate enough to use equilibrium states at all, then the potentials do not feel the change rate.

*Exceptional situations:* The above argumentation may be invalid if some physical laws or the geometry are explicitly time-dependent. However, then there are problems with the formulation of thermodynamics anyway [19], and with equilibrium.

*Comment:* CPT symmetry can be derived from Lorentz symmetry + causality. So its existence is doubtful in a time-dependent Universe. However, up to now no reliable tratment of quantum fields exists in a geometry without a temporal Killing vector, therefore it is difficult to define the symmetries of quantum fields in the early Universe.

In addition we formulate a purely phenomenological and microscopy-free

*Theorem 4:* Cond. (5.5) can be simultaneously fulfilled for all macroscopic systems composed only of particle R if particle-free states of any of such systems hold chemical equilibrium only with particle-free states of others.

*Proof:* Note that the physical existence of such a system was not involved; we are free to work with all systems, real, hypothetical or unknown, whose entropy function fulfils the conditions usual for S(V,E,N).

Consider a system with even entropy function S_{(1)}(V,E,N)=S_{(1)}(V,E,-N). For such a system

(5.7) µ_{(1)}(V,E,N=0) = 0

because of the symmetry, i.e. System 1 fulfils Cond. (5.5) in a special convention. Now, take any second system from the set involved in the theorem. According to our assumption and eq. (5.7):

(5.8) µ_{(1)}(E_{(1)}/V_{(1)},0) = µ_{(2)}(E_{(2)}/V_{(2)},0) = 0

Hence ¶S/¶N=0 at N=0, that is, in the same convention Cond. (5.5) holds for System 2 too. And so on.

Here we stop. Our conclusion is, then, that in a wide class of systems (of reasonable behaviour) of a *single* kind of elementary particles there is at least one thermodynamic convention in which Cond. (5.5) simultaneously holds for all of them. Since the possibilities for Conds. (3.1-2) have already been discussed, we can conclude as follows:

All macroscopic systems in equilibrium, with a single kind of particle R (together with its antiparticle R_{anti}), which obey the Callen axioms and CPT symmetry, can be classified into 2 disjoint classes as follow:

1) those where Conds. (3.1), (3.2) and (5.5) can be simultaneously fulfilled in a convention

**C**°; and

2) those in which the limit involved in Cond. (3.2) does not exist.

Then we can define the "special" convention **C**° via the systems of Class 1) (if it is not empty, which we now assume) and then it is uniquely fixed except for K of the temperature unit. Since for the systems of Class 2) the construction is impossible, they do not define **C**° otherwise.

So far about "elementary" particles. But many systems considered in thermodynamics contains composite entities as particles. And observe that the above Theorems do not help for "composite" particles, e.g. for nonrelativistic macroscopic bodies as billiard balls, (except if they obey CPT symmetry for some reason of their own). This will be shown in details in Sect. 7, here we make only two brief comments.

First about compositeness. We restrain ourselves here from clear-cut definitions. A proton is generally treated as an elementary particle; still by any probability it is composed of 3 quarks. Any particle now believed to be elementary may turn to be composite later. However in the present context we call a particle "elementary" if it may participate in pair creation and annihilation.

This is not a matter of stability. Neutrons are unstable but neutron-antineutron pairs can be produced. In principle anti billiard balls could be created from antiivory, consisting of antiprotons, antineutrons and positrons; and then a billiard ball and an antiball can annihilate. However it is hopeless to pair create billiard balls, since the internal structure could not survive at the energies necessary in the pair creation. So one can expect situations at T ~ Mc² when an "elementary" particle distribution contains more or less equilibrium number of pairs, but cannot expect such for billiard balls. True, some hadrons are at the verge of the "elementary" behaviour in this sense, because the energy needed for creating resonances as excited states is in the same order of magnitude as needed for pair production.

Therefore in equilibrium thermodynamics anti billiard balls or anti protein molecules are practically particle kinds completely independent of billiard balls or protein molecules. This means that C symmetry cannot be expected for the phenomenologic S function, so the Theorems do not help; in the practical situations nothing guarantees that S have an extremum at N=0. Then one can fix the other two zero points, but afterwards

(5.9) lim_{n->0} ¶s/¶n = n(e)

Were this n constant, still Transformation (2.5) could bring it to 0, so in the phenomenologic thermodynamics of the given composite "particle" the zero point of µ/T could be fixed according to the scheme suggested. However n(e) is generally not constant; e.g. for Boltzmann gases it takes the form -mc²/T

Of course one may try with a double limit. If V is fixed, N -> 0 and E -> 0 or ¥, then n is a constant which can be removed by the remaining transformation. However -mc²/T diverges at E->0, so the removed constant would be infinite creating ambiguities. The E -> ¥ limit may be free of this problem. This point deserves further elaboration, which however will not be done in the present paper; but a composite body may not survive this limit.

Let us repeat that the zero point fixing is necessary for each kind of particles *separately*, because each existing, possible or imaginary object which may be a particle in a thermodynamic system is an individual N^{I} on its own right. Therefore Transformation (2.5) works on S separately according to each N^{I}; there are as many different C_{I}-s as the number of possible particle kinds. If one is building up the thermodynamics of billiard balls, then in that formalism the billiard balls are particles on their own right and their chemical potential needs a zero point fixing.

Still one can work in a way similar to that for systems with limiting temperature. There we fixed the zero point of temperature for the "generic" systems, and then in equilibrium of "generic" and "exceptional" systems the temperature scale of the latter ones is fixed too. Now we may fix first the zero points for "elementary" particles, and then in systems containing a composite "particle" and its "elementary" constituents as well the chemical potential of the composite one will be fixed too. There remain problems with composite particles whose composition is not known or which never were in equilibrium with anything else. The zero points of such systems are not fixed operatively by the present condition.

6. THE RELATIONSHIP OF C° AND C'

Now, at least for equilibrium systems with a single kind of particle, with CPT symmetry and fulfilling the Callen axioms, we are in the position to compare the conventions **C**° "simplest in thermodynamic sense" and **C**' "nearest to mechanics". Let us repeat how these two conventions are defined. In **C**° the entropy functions of "generic" (non-exceptional) systems obey the simple conditions

lim _{E->0} S = 0

(6.1) lim _{E->¥} ¶S/¶E = 0

lim _{V->¥} ¶S/¶V = 0

lim _{N->0} ¶S/¶N = 0

Then from **C**° a general convention **C** is reached by the general permitted transformation

(6.2) S -> S +AV +BE + CN

and then any convention **C**, relative to **C**°, is defined by the triad of constants:

(6.3) **C** = **C**(A,B,C)

Now, **C**' is the convention in which the *thermodynamic* pressure

(6.4) p(**C**') = (¶S(**C**')/¶V)/(¶S(**C**')/¶E)

is "as near to" the *dynamical* pressure P of mechanics "as possible".

In usual "local" mechanics the zero point of P cannot be measured, only its gradients. However, in General Relativity P (i.e. the average spatial diagonal component of the energy-momentum tensor) directly influences the curvature, so, having the curvature measured, P can be determined backwards. Then we can choose an appropriate system, determine its P from observation, and compare it to p, determined by pure thermodynamics. By this way we *measure* (A,B,C) of **C**'. Afterwards if one formulates a statement in **C**°, it can easily and uniquely reformulated in **C**' and vice versa. Our appropriate system, where P will be measured and p(A',B',C') compared to it, will be the Universe.

We choose the Universe for various reasons. First, it is the only system without exterior, so the only one for which no external disturbing effects are possible. Second, it is the only system of which all other systems, thermodynamical or not, are subsystems, in contact with it. Third, at least in the present Universe, the dynamical pressure P possesses "almost" the same variables as p, which is convenient for comparison.

The first two statements do not need explanation; let us clarify the third. In the presence of transport processes P cannot be equal to p, since the first one includes transport contributions, depending on such non-thermodynamic variables as velocity and temperature derivatives &c. In the simplest near-equilibrium case there are at least 3 such contributions, from shear and bulk viscosities and heat conduction. Symbolically we can write

P = p*(e,n) + (2h(e,n)+3h'(e,n))u^{r}_{;r} + q^{r}_{;r}

(6.5) q^{i} = -k(e,n)T,_{r}(g^{ir}+u^{i}u^{r})

Here p* may or may not be equal to p(**C**), but at least has the same variables. The above condition "be as near as possible" can mean only

(6.6) p(**C**') = p*

Now, in some physical systems the transport contributions can be substantial. However in the Universe shear viscosity and heat transfer are impossible (on large scale) due to the cosmological principle (homogeneity + isotropy), and in the present state the bulk viscosity is negligible [20].

So let us observe the Universe, determine **C**', and try to measure A', B' and C' belonging to this convention. Here we refer some results of [2].

Up to now there are no evidences that **C**° and **C**' would differ at all. The available (scant) *bounds* come from heavy ion physics and cosmology.

1) Eq. (2.5) together with the Callen axioms [5] leads to

(6.7) B ³ 0

otherwise S would not be monotonous with E. Although one may alter the axioms, now we accept this bound.

2) The cosmological calculations for the primordial nucleosynthesis give essentially good results. So **C**° gives roughly the cosmologic (**C**') T at 1 MeV [2]. In heavy ion calculations both hadrochemical and dynamical results are not too bad up to T=150 MeV, and heavy ion physics uses p for P. Therefore probably

(6.8) B < 1/(100 MeV)

3) In the Friedmann equations of cosmology the absolute pressure appears. In the present Universe the matter physically seems to be in first approximation an ideal gas and no effect of the constant A is seen [2]. Therefore roughly

(6.9) -10^{-6} cm^{-3} < A < +10^{+3} cm^{-3}

This means that up to now there is no evidence against **C**°=**C**'. In Ref. 2 we proposed to set the zero points by the Universe itself, which is the most general of all systems. However, still the C_{I}'s remained free.

There is not absolutely impossible to fix some C_{I}'s in a similar manner. Namely, the change of a C_{I} alters the reversibility of processes in which N^{I} changes. However, for a *conserved* particle the only effect is an additive constant in the entropy, which is irrelevant. So up to now no bound has been found in cosmology for the constant C. C of **C**' may be 0; direct observations are in principle possible for some lepton and baryon fields, but for quarks only indirect data, reconstructed for the early Universe, can be used. The most which can be said is that there is no evidence against C_{I} in **C**'. However, we will return to this problem in Sect. 9 on the grounds of the formal simplicity.

7. MORE THAN ONE PARTICLE COMPONENTS

For more than one independent particle components the simple results of the previous Chapter do not follow. This will be shown here by using *two* components. In addition, for simplicity's sake, we assume C symmetry. Then

(7.1) S(V,E,-N^{1},-N^{2}) = S(V,E,N^{1},N^{2})

But now this condition permits a more general form than in Chapter 7. To see this, let us introduce 3 redundant but C-invariant particle variables as

x º N^{1}²

(7.2) y º N^{2}²

z º N^{1}N^{2}

Then the C-symmetry simply requires

(7.3) S = S(V,E,x,y,z)

where the x,y,z dependence is arbitrary (but, let us say, smooth, analytic, &c).

The chemical potentials can be expressed as

(7.4) µ_{1} = 2S,_{x}N^{1} + S,_{z}N^{2}

µ_{2} = 2S,_{y}N^{2} + S,_{z}N^{1}

Now, consider the N^{1}=0 hypersurface in the 4 dimensional state space. There

(7.5) µ_{1}(N^{1}=0) = S,_{z}N^{2}

which is generally nonzero. Clearly S,_{z} is an interaction term. Since at the hypersurface N^{1}=0 z changes its sign, there is no general reason for S,_{z}(z=0)=0. So µ_{1}(N^{1}=0) does not follow from C symmetry. The freedom (2.5) does not help, since it cannot introduce or remove an interaction term. (The linear terms can be exploited for achieving even functional forms for S around N^{1}=0 and N^{2}=0.) If so, then the zero point fixing becomes complicated for more than one particles. The complication is subtile enough for further works. However here we at least outline the possible way of solution. We emphasize that at this point we deal still with "elementary" particles without at least obvious internal structures and obeying fixed laws.

We assume that the system investigated contains two and only two kinds of elementary particles, interacting via elementary laws, *including* *C* *symmetry*. In addition, we assume also that the system is *not* exceptional in E and V. Then

lim_{E->¥} ¶S/¶E = 0

lim_{V->¥} ¶S/¶V = 0

(7.6) lim_{N1->0} ¶S/¶N^{1} = n_{1}(E/V,N^{2}/V)

lim_{N2->0} ¶S/¶N^{2} = n_{2}(E/V,N^{1}/V)

The n's come from interactions, and the interactions come from elementary laws, so they cannot be switched off. However, from C symmetry at least

n_{1}(V,E,N^{2}=0) = 0

(7.7) n_{2}(V,E,N^{1}=0) = 0

in the same way as for a single particle. So the double limit N^{1},N^{2}->0 can fix C_{1} and C_{2}. (To see this, cf. eqs. (7.1-5).) Therefore if our assumptions hold but eqs. (7.7) do not, then the only possibility is that we are *not* in **C**°. So **C**° has been found if

lim_{E->¥} ¶S/¶E = 0

lim_{V->¥} ¶S/¶V = 0

(7.8) lim_{N1,N2->0} ¶S/¶N^{1} = 0

lim_{N1,N2->0} ¶S/¶N^{2} = 0

We ignore the exceptional systems.

Very briefly we discuss transmutations. With only two particle components the simplest possible transmutation law is

N^{1} <--> N^{2}

as e.g. for nucleons and D resonances. Then, in equilibrium

(7.9) µ_{1} = µ_{2}

(For more particles a number of such linear relations follow with stochiometric numbers in them.)

Now it is easy to see that eq. (7.10) is compatible with the zero point fixing above. Namely, when both particle numbers are 0, both chemical potentials are 0 too.

8. ON COMPOSITE "PARTICLES"

Now we return to thermodynamic systems of composite particles and phenomenologic laws. Then, as we saw in Sect. 5, theorems about C, CP, or CPT symmetry of fundamental laws not necessarily help. Therefore µ/T may not be constant in the N->0 limit, and such a term cannot be removed by Transformation (2.5).

As an example consider billiard balls. A billiard ball is a composite entity of very specially arranged elementary particles, which have CP symmetry (kaons ignored).

In a world composed only of billiard balls and nothing else, µ_{BB}(N^{BB}=0) would follow from C symmetry for billiard balls (assumed). However in our world at energies far below those needed for creating anti billiard balls, the balls start to be decomposed. So there are two possible treatments.

Case 1: Very low energy

Well below the chemical energy scales in ivory billiard balls are indesctructible and indivisible particles. Then no anti billiard balls are available, N^{BB}³0, and no C symmetry can be utilized.

Let us fix the zero points of 1/T and p/T in the usual way. We are not yet in **C**o, because addition of CBBN^{BB} to S is possible. Evaluate S in the usual way in any definite member of this subset of **C**'s. Then

lim_{N(BB)->0} ¶S/¶N^{(BB)} º

(8.1) -(1/T)lim_{N(BB)->0} µ_{BB} = n_{BB}(E/V)

If this empirically determined n_{BB} is a constant, then it can be removed by means of Transformation (2.5), and then we have arrived at **C**°. If n_{BB} turns out to be e-dependent, then we cannot directly find **C**°.

We might try with a double limit lim_{E->(0 or ¥)} n_{1(BB)}. By construction that limiting value is a constant. However, as told in Sect. 5, for E->0 n_{BB} is probably infinite. There would be arguments for the finite limiting value in E->¥; however in that limit the billiard balls decay, and the system cannot be investigated.

Case 2: Unrestricted energies

Going up with energies the billiard balls decompose. First ivory powder appears

BB<-->IP

and in equilibrium

(8.2) ¶S/¶N^{(BB)} = M*¶S/¶N^{(IP)}

where M is roughly a quantity of ivory powder equalling a ball. But the tooth tissue of elephants is also a complicated arrangement of atoms, whose laws are completely empirical.

At high enough energies the disintegration goes further to "elementary" particles. So

BB <--> IM

IM <--> {IM} + {EP}

where {IM} stands for the whole set of entities intermediate between complete billiard balls and Elementary Particles. Now this is a multiparticle system

(8.3) S = S(E,V,BB,{IM},{EP})

with transmutations. The result is a set of chemical equilibrium equations

(8.4) k^{ßBB}µ_{BB} + k^{ß{IM}}µ_{{IM}} + k^{ß{EP}}µ_{{EP}}=0

with stochiometric numbers k in the reaction channels ß.

Now these equations connect the chemical potential of billiard balls to those of elementary particles. Elementary particles can be investigated at temperatures well above the domain of existence of billiard balls. Therefore there are situations with only elementary particles present, and then the zero points of their chemical potentials can be fixed on the way mentioned in Sect. 5. Then, via eq. (9.4), the zero points of µ{IM} and µ{BB} have been fixed, even if the corresponding composite bodies do not obey C symmetry. At lower temperatures via eq. (8.4) the convention **C**° can be found even for low energy systems of mere billiard balls.

For composite bodies of unknown composition this process is impossible. However then clearly our information about the bodies is incomplete. Unfortunately for them **C**° cannot be found in that state of knowledge.

9. ON THE THERMODYNAMICS OF THE EARLY UNIVERSE

We close the paper with a very superficial discussion of the thermodynamics of the early Universe prior to formation of atoms. We do this to see the connection between **C**° and **C**' (**C**^{U}) in the particle degrees of freedom.

According to present knowledge, before 10^{-36} s, i.e. above 10^{15} GeV/particle, the matter seems to have been in the symmetric phase of the Grand Unification Theory. In simplest GUT the group of fundamental interactions is SU(5); the "ultimate" elementary particles are [21]:

i) spin 1/2 fermions, i.e. quarks and leptons;

ii) spin 1 bosons mediating interactions, i.e. X and Y (leptoquarks), gluons, W and Z bosons (including the precursor of photon);

iii) spin 0 bosons of self-interactions (Higgs bosons).

In the symmetric phase all fermions behave similarly, and all vector bosons have the same coupling strengths, while the expectation values of all scalar bosons are 0. The independent particle components are the various fermions.

The SU(5) theory contains only 2 particle conservation laws:

Z = const. (electric charge)

B-L = 3Q-L = const. (baryon N° - lepton N°)

All the particle ratios not affected by these conservations were redistributed repeatedly by transmutations according to equipartitions, and the vector boson expectation values were determined by E/V, Z/V and (B-L)/V.

The Universe is unique, so its Z and B-L are unique values. We do not know of any structure formation in the symmetric phase of GUT, so probably permanent inhomogeneities did not exist; fluctuations were correlated within l~~~h~~c/T.

Since Z and B-L are conserved in any spontaneously symmetry-broken states of SU(5) GUT, these values can be measured in the present Universe, even if great caution is needed because of uncertainties and of the danger in extending local data to the whole Universe.

In our neighbourhood the matter seems neutral on large scale; no astronomical observation shows electrically charged stars, galaxies or galaxy clusters. As for the whole Universe one can proceed as follows. The so called "Copernican principle" is often meant as suggesting isotropy + homogeneity. Isotropy can be checked on the 2.7 K blackbody radiation, which is indeed isotropic in temperature up to 5 digits. The 10^{-5} changes can be explained by e.g. spontaneous thermal fluctuations during primordial inflation at the breakup of the GUT symmetry [22]. As for homogeneity galaxy counts definitely show grouping or "walls" separated by cca. 140 Mpc [23]. So there is no homogeneity on this scale. However this structure seems roughly periodic [24], in which case on larger scales there is no evidence against homogeneity.

Then let us accept maximal spatial symmetry (6 symmetries), as generally done in theoretical cosmology. The only possibilities for the metric tensor with so many symmetries are [25]

ds² = dt² - R²(t){dx²+f²(x)dW²}

sin x (k=+1; SO(4))

(9.1) f(x) = x (k=0; E(3))

sh x (k=-1; SO(3,1))

Now, a noncompensated electric charge would result in a global electromagnetic potential Ai. But the only vector potential with one of the above symmeties is

(9.2) A^{i}(x^{k}) = (A^{o}(t),**0**)

which form leads to vanishing field tensor. Therefore a homogeneous isotropic Universe would not be possible with homogeneous noncompensated electric charge. Therefore it is better to accept Z=0.

For B-L we can use particle counts, although for some particles it is practically impossible and the data are rather local. In our neighbourhood the overwhelming majority of fermions are protons, neutrons, electrons, neutrinos and antineutrinos. The counts show

n(p) = n(e) (electric neutrality)

n(n) » 0.16*n(p) (from stellar He)

and neutrino counts are still impossible. However

B/L » (n(p)+n(n))/(n(e)+n(n)-n(n_{anti}))

is in the order of unity if (n(n)-n(n_{anti}))/n(e) is in the order of unity, which is not impossible. So present data do not rule out

B-L = 0.

In this case all the conserved quantum numbers of the very early Universe would be those of the vacuum.

Then during the symmetric phase of GUT the thermodynamic description needed an entropy function

(9.3) S = Vs(e=E/V)

with a form similar to that of the radiation fields. In addition, near to Planck time and/or energy a further extensive may have been necessary for the cosmologic Hawking radiation [19].

The spontaneous breakdown of GUT symmetry is extensively discussed in the literature. Here it is sufficient to mention that the endproduct is the "standard theory", SU(3)xSU(2)xU(1). In this theory baryon and lepton numbers are independently conserved. In the simplest case this would mean

(9.4) S=Vs(E/V,B/V,L/V)

(According to experiments, the lepton conservation holds in 3 different families separately, which fact now will be ignored for simplicity.) Since baryons and leptons interact, we are back at the situation of Sect. 7. (CP is violated in the SU(2) sector, but CPT may be exact or a good approximation in a sufficiently old Universe.) Now for particle numbers **C**U is defined so that formulae be simplest. Note that interactions in the B gas include QCD, while in the L gas or between the two ones do not, so they are weaker. Since QCD has asymptotic freedom, in first approximation even in **C**^{U} (**C**') one gets

(9.5) µ_{B}(V,E,B=0,L) = 0

µ_{L}(V,E,B,L=0) = 0

Up to now no evidence is known in this era for differences between **C**° and **C**^{U}, i.e. for nonzero C_{B} and C_{L}; and for any case they are bound from above by the strengths of non-SU(3) interactions.

However this simple picture becomes rather complicated after the quark confinement transition (rougly between 7 and 12 µs, at 160 MeV temperature [26]). After hadronisation the SU(3) sector is represented by a great diversity of strongly interacting hadrons. Then the interactions may introduce a number of C_{I}'s between **C**° of eq. (7.8) and **C**^{U} of maximal simplicity of formulae. Still, all these interactions go back to 3 fundamental ones, so the C_{I}'s should be expressable with 3 independent coefficients, but the description of hadronic interactions is not yet in this stage.

At the end of the plasma era (t~300000 y, T~3000 K) atoms are formed, then molecules and so on. Thenceforth the world contains very complicated entities as e.g. billiard balls, for which only phenomenologic descriptions are available. Then we are inevitably back to the problems of Sect. 8, but here we stop.

10. CONCLUSIONS

The present paper is only the first attempt to fix the zero points of the chemical potentials in a pure thermodynamic way. The problem may be arbitrarily complicated for complicated systems. However here we at least have shown that

1) by observing the behaviours of thermodynamic systems the zero points of -µ_{I}/T cannot be determined;

2) simple thermodynamic principles can be formulated for these zero points;

3) these principles can simultaneously be obeyed by various independent elementary particles with C symmetry;

4) and the zero points of the above elementary particles will determine the zero points for particles composed of them.

However, not all the elementary particles (fields) possess C symmetry. For example. weak interaction violates it. It seems that without spatial ordering CP symmetry suffices, and, in equilibrium, CPT is enough. Since CP does not hold for hyperweak interaction, and CPT is doubtful in the very early Universe, systems of kaons deserve further attention.

It seems that for elementary particles the special thermodynamic convention **C**° of the zero point principles is at least near to the "naïve" ones where dynamical and quantum field calculations generally happen.

ACKNOWLEDGEMENT

The author acknowledges that the zero point condition for the chemical potential, used in this paper, had been a result of discussions with Dr. K. Martinás, and originally the paper was planned to be written together with her; however she has declined.

***

The work has been partly supported by OTKA Funds N° 0740 & 014304.

APPENDIX: THE ZERO POINT CONSTANTS IN THE STATISTICAL APPROACH

Here we show that the freedom (2.5) appears in the "microscopic" statistical treatment as well, therefore it is not a consequence of any loss of information in macroscopical averaging. For simplicity consider a system with a single conserved particle component and with negligible two-particle correlations.

If so, then all the information is carried by the one-particle momentum distribution function f(**p**;**x**,t). Again for simplicity assume spatial homogeneity. Then, by definition,

(A.1) N = Vòf(**p**)d**P**

(A.2) E = Vòe(**p**)f(**p**)d**P**

where e is the one-particle energy and d**P** is the momentum volume element, either for nonrelativistic or for relativistic case.

The entropy is a quantity proportional to volume, which is constant for reversible changes and increases for irreversible ones. Therefore

(A.3) S = -V**H**

where **H** is a functional of f, whose change is negative semidefinite ("Boltzmann's H-function".) For example, for a nonrelativistic gas of distinguishable particles obeying the Boltzmann equation such a functional is

(A.4) **H**° = òf(**p**)lnf(**p**)d**P**

However, if **H**° has negative semidefinite production, then any other functional of form (A.4) with the new integrand

(A.5) flnf -> K²flnf - A - Bef - Cf

will have negative semidefinite one as well, since the contribution of the first new term is constant by construction, and that of the second and the third by energy and particle conservation in collisions. Via eqs. (A.1-4) one gets just the freedom (2.5).

The same result is obtained by observing fluctuations. Since eS is proportional to the state probability, in lowest order approximation the probabilities depend on second derivatives of S [27] (for higher terms see Ref. 26), therefore in fluctuation formulae A, B and C drop out, while K² scales with the definition of unit volume. Therefore one arrives at an unpublished observation of Martinás [28]: namely that, if instead of thermodynamic measurements on many systems one observed one system at different particular states *together with its fluctuations*, then the entropy function could again be determined up to the freedom (2.5) and just the Riemannian structure would again be uniquely obtained.

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