a proposed principle of thermodynamical equilibrium and the classical theory of the black body

IL NUOVO CIMENTO VOL. 105 B, N. 12 Dicembre 1990

A Proposed Principle of Thermodynamical Equilibrium and the Classical Theory of the Black Body.

G. MASTROCINQUE

Dipartimento di Scienze Fisiche, Facolt~ di Ingegneria dell'Universitd - Napoli, Italia

(ricevuto il 28 Aprile 1987; manoscritto revisionato ricevuto il 2 Gennaio 1989)

Summary . - - In a previous paper, we developed a so-called fluctuation model for the inelastic collisional interactions gove~aing molecular energy transfers. The model results in a procedure to be applied to the classical equations for energy transfer probabilities in order to obtain (,symmetrized)~ expressions respecting the detailed balance principle. Since the procedure reveals some generality, we revisit in this paper the classical radiation problem, i.e. the interaction between the classical electromagnetic field and the black body. We propose a (,classical alternative~ to Planck's procedures. Based on the ,,continuum)~ physics assisted by appropriate probabilistic postulates, our method seems able to take into account energy ,,fiuctuations~ hE and to result in the correct Bose-Einstein or Planck energy distributions for the radiation and the mechanical oscillator in thermodynamical equilibrium, inclusive of a zero-point energy term. We propose the procedures for consideration in this paper.

PACS 05.70 - Thermodynamics.

1. - I n t r o d u c t i o n .

In a previous paper [1] we introduced the so-called classical fluctuation model for inelastic molecular collisions. We showed that this model brings the classical equations for energy transfer probabilities to respect the detailed balance principle (DBP) and to provide results equivalent to quantum-mechanical (QM) ones in standard approximations (WKB, SPA). The model relies on introducing, with the help of classical mechanics, a function hSc (El, El) called ,(collisionab, or (Snteractive~) entropy (difference), symmetrically dependent on the initial E i and final Ef energy values of the states characterizing the inelastic-collision channel.

Our model and entropy definitions in [1] are limited by assessed approximations, but they provide a consistent framework to investigate a number of classical problems in physics. For instance, we showed that the so-called classical-discrete fluctuation model results in an interesting classical alternative [2] to the QM Jackson and Mott equation for vibration-translation energy transfers in gases governed by an exponential-repulsive intermolecular potential. We also showed [3] that an important

1315

1316 G. MASTROCINQUE

improvement can be achieved in the classical Bethe calculation for the energy loss and radiative cross-sections of charged particles in stopping media. In order to apply the model to the cases of ionic collisions and of Coulomb excitation of nuclei further work is in progress.

In this paper, we investigate the fundamental case of the black-body radiation and propose a physical model for the mechanical oscillator. On the basis of our method, we propose a new <,classical alternative,, to Planck's procedures to state the energy distribution equation for the radiative field interacting with the mechanical oscillator and for the oscillator itself. Nernst [4, 5, 6] since 1916 first proposed classical models for mechanical oscillators resulting in the correct Bose-Einstein distribution. His models are based on introducing a <<stochastical>, energy threshold to be overcome in order that the oscillators are allowed to exercise thermal exchanges in Boltzmann's sense.

We believe that a unified point of view could be developed where both our ideas and those by Nernst coexist, but we point out that our method is different from the Nernst one. It is essentially concerned with the concept that the internal energies of two interacting physical systems, due to energy exchanges, fluctuate between two levels. Even this concept is not new. Although in a different form, it was already considered by Planck in his so-called second theory [7]. However, by means of our model, we provide a physical and mathematical framework allowing classical physics to take into account the energy exchanges or fluctuations. This is simpl~} achieved by <,symmetrizing,,, as showed in[l], the classical equations for the energy transfer probabilities.

In order to set up a consistent procedure to investigate the black-body case, we also propose in this paper to consider what we think is a fundamental principle of the thermodynamical equilibrium for a closed collisional system. This principle can be expressed by the equality between the corresponding collision-induced entropy difference ASc between states and the thermodynamical entropy difference AST governing the thermodynamical fluctuations.

We give examples of utility of this statement. Although we do not investigate further, we think that such a property should have a deep relationship with the ergodicity properties. It is indeed based on the belief that the values of quantities calculated by means of mechanical, deterministic laws--when an appropriate probabilistic postulate is stated will turn out to be equal to the corresponding <<statistical ensemble,, mean values.

The fluctuation model is based on the assumption of a continuum energy spectrum available to the oscillators, and the method we propose here seems able to give the correct Ptanck or Bose-Einstein distribution laws for the black-body radiation and the mechanical oscillator, including the zero-point energy contribution. If our procedures are accepted or improved, we believe that they might appear to lead the classical physics to overcome the Rayleigh-Jeans law and to perform a fundamental step towards unification with quantum physics.

2 . - B a c k g r o u n d .

Both to the purposes of comparison and to set up the theoretical framework in sect. 3, based on the chemical equilibrium theory, we give here some fundamentals concerning a) the fluctuation model, b) the chemical equilibrium constant, c) the

A PROPOSED PRINCIPLE OF THERMODYNAMICAL EQUILIBRIUM ETC. 1317

classical thermodynamic model for the emission/absorption of radiation by a mechanical oscillator.

2"1. The fluctuation model. - As proved in [1], improvements are obtained in the classical expressions for the energy transfer probabilities in molecular collisions by ~,symmetrizing~ the expressions themselves according to the following rule:

(1)

L E~

In eq. (1), Pc (E) is the (classical) probability that an energy amount AE = E f - Ei is transferred from one degree of freedom to another due to an inelastic process occurring with collision energy E. According to the perturbation theory, E is generally assumed to be a constant when performing classical calculations even for inelastic processes. This assumption can be overcome by means of the fluctuation model. The improved expression for the energy transfer probability, given by eq. (1), is obtained on the contrary assumption that E fluctuates during the process between the extreme values Ei, El. So Pif(Ei,Ef) is called a ~<fluctuation- transition probability. The quantities

(2) ASe (E) = In Pc (E),

(3)

Ef

1 IlnPc(E) dE ASc (Ei, Ef) -- ~ Ei

are called <<collisionab~ or ~,mechanicab~ entropy (differences), introduced to take into account the disorder in the relevant energy level populations due to the inelastic collision.

We introduced our model and equations in[l] mainly referring, as a primary example, to the case of an intermolecular vibration-translation energy transfer and to the hypothesis of ,,smalb~ probabilities. However, we point out here that the resulting symmetrization rule expressed by eq. (1) can be interpreted in the frame of the fluctuation model and generalized to many different cases, independently of the previous assumptions. On the basis of our investigations we definitely assume that eq. (1) can be generally applied to various processes in physics.

For the purposes of this paper, we note here that rule (1) can be equally applied to the products or ratios of suitable probability functions. As far as we can identify--as it will be shown in the next sections--the chemical equilibrium constants Ke with the ratios of probability functions appropriate to the forward (f) and backward (b) processes at hand, we can introduce the following symmetrization rule for Ke:

(4) Ke(E) ~ Ke (Ei, El) = exp lnKe(E) dE .

E1 J

1 3 1 8 G. M A S T R O C I N Q U E

We will also use in the sequel the following quantities:

(5) ~h~ e (E) = In K e (E),

Ef

1 f l n K e ( E ) d E (6) ASe (Ei, El) = - ~ .

Ei

These quantities are, in the quoted approximations, the appropriate entropy functions useful to take into account the energy fluctuations occurring during a chemical process.

2"2. The chemical equilibrium constant. - The rate constants K of a chemical process are generally given in two standard forms. The first one is the Arrhenius equation:

(7) KA -- AQ = ZPQ -- ZP exp - - - ~ .

In eq. (7), A is a frequency factor given by a ,number of collisions,, Z times a probability factor P. E -+ is the activation energy of the transition complex, kT is the thermodynamical Boltzmann energy and we define the factor Q as the Arrhenius exp [ - E -+/kTl.

The second standard form for K is the Winne-Jones and Eyring expression:

= -~--exp - - ~ = T

In this equation, h is the Planck constant, H • is an activation enthalpy and AS -+ is an activation entropy given by

(9) aS -+ ---- S a c t . s t a t e - S r e a e t a n t s .

Iterating the Arrhenius equation (7) for the forward (f) and backward (b) processes we have

(10) K ~ - K A ( T ) - Z ~ P f Q f Pf [ A E • Kb ZbPb Q----~ - Pb exp ~ .

To write the last term of this equation, we took for the purposes of this paper:

(11) Zf = Zb,

(12) E~ - E~ = AE •

Iterating the Winne-Jones/Eyring expression we have

(13) K f = KW~E (T) = exp [Sf - Si ] . Kb

To write this equation, we purposely took (see eqs. (19), (22), (23), (31),):

(14) H~ - Hf • -- O.


We also have

(15) Sf - S i -- Sprod.ucts - Sreactants �9

Sf, S~ are the relevant entropies for the final and initial states of the chemical system, respectively. We will use in this paper both the expressions (10) and (13). We also will use the (,mass action law, which can be written, in agreement with customary models,

(16) Kf = K(T) - N~NI gb NiN2"

This equation is appropriate in the frame of the ensemble.

classical ,~grand-canonical-

According to the simplest expression of the mass action law we assess with eq. (16) that in the thermodynamic model two reactants in appropriate concentrations Ni, N2 proceed along the reaction or the inelastic process path towards the final states and products. These products will have concentrations equal to Nf and N1, respectively.

2"3. Thermodynamic model of the emission~absorption of radiation by a mechanical oscillator. - Let us introduce here a thermodynamic model relevant to the classical radiation problem, i.e. the interaction of a classical (unidimensional single-mode) electromagnetic field (CEF) and a classical (harmonic, linear) mechanical oscillator (CM0; sometimes, CH0-when harmonicity is the specific property involved in the context). CEF and CM0 have a continuous energy spectrum, constant ~,constant volume specific heats~ and are supposed able to store/exchange an energy amount AE via a resonant process. The emission/absorption of radiation by CM0 is regarded here as a chemical process.

We start conceptually the process activating an electrical dipole charge for CM0 and by ~,injecting, an ~out-of-equilibrium, energy amount AE into the classical, unperturbed thermodynamic system composed by CEF + CM0. Then the systems go out of the initial equilibrium state and proceed towards another equilibrium state, where the energy AE is dynamically shared via emission/absorption of radiation.

In our view, the final equilibrium state is given by a situation where the two systems continuously ~fluctuate~ between two extreme out-of-equilibrium states, running over the so-called thermodynamic path--/.e, the ensemble of equilibrium states relevant to the unperturbed systems. Let E i , Efbe the CEF energies relevant to the extreme fluctuation states, with Ef I> E i by convention and E l - Ei = AE. Let E~, E~ be the corresponding CM0 energies relevant to the extreme fluctuation states, with E~ >/E~ by convention and E~ - E~" = hE. The initial out-of-equilibrium situation can be illustrated as follows. Suppose the CEF mode starts with N ~ field carriers at the energy E i and N o field carriers at the energy El. Suppose CM0 starts with N ~ carriers at the energy E~' and N1 ~ at the energy E~. The total energy of this system is

(17) E T = Ni~ + N~ N ~ + N ~ = 2kTi + h E .

In this equation, Ti is the initial absolute temperature both featuring the unperturbed classical systems. To be consistent, we take, in the same equation,

(18) N ~ o = Y ~ ~ = N ~ ~ = Yi ~ o = 1.


Then we find

(19) ET = E l + E * + ~ = E i + E ~ = Ef+ E~ = E + E * = const.

Here E and E* are the current values of CEF and CMO thermodynamical energies, as better defined in the following eqs. (20), (21). The initial out-of-equilibrium situation is to evolve towards the final equilibrium state characterized by eq. (16). In that equation, we identify by convention the Ni CEF carriers and N2 CM0s as the (,reactants~. Nf CEF carriers and N 1 CM0s then are the reaction ,,products)). We will also identify the N2 class of CMOs as the ,~emitters)~ and the N1 as the (~absorbers)). Again, as a matter of convention, the ,,forward,) process will be here the emission of the energy amount hE by CMO. The ,(backward~) process is the absorption.

Finding for the case at hand by the following proposed procedures the expression of the equilibrium constant K(T), the equilibrium values of ,~reactants)~ and ~products,) concentrations as well as the thermodynamical equilibrium energies of Ni, Nf, N1 and N2 classes of CEF, CMO carriers is the purpose of this paper.

CEF and CMO are classical systems, each displaying two degrees of freedom. Since the CEF energy is a linear function of its momentum, we take equation (20) for the (unperturbed) thermodynamical function ~CEF. In the uni-dimensional model we refer to, this is orthodox for a continuous energy spectrum. As far as the mechanical oscillator is regarded, it is well known that setting up an orthodox expression for its thermodynamical function--consistent with (quantum) experimental evidence reveal- ed in the last century a very diffcult, if not impossible, task to accomplish for the classical physicist [7]. In order to display our procedures, however, we will assume the following equation (21). This last agrees with the following considerations:

i) By the Madelung formulation of the SchrSdinger equation, we know that any classical interpretation of quantum effects is, in turn, a hydrodynamic theory. Since the classical Bernoulli invariant is an energy per unit volume, instead of the standard - k l n T* we take for ~cMo the expression - k l n T * / V * .

ii) This assumed expression is also consistent with the fact that, contrary to the case of perfect gases, for linked systems one finds that adiabatically increasing the volume (i.e., the oscillation amplitude) makes the energy to increase (,(negative pressure~, is involved in calculations).

Because of ii) we note that it is convenient to change sign to the standard convention for the thermodynamic constitutive equation of the CMO pressure, p*. Therefore, we will take this last as a positive quantity, with the consequence that standard conventions concerning work and enthalpy will also be reversed in the following.

With these assumptions, we have for the. unperturbed thermodynamic functions:

(20) E ~CEF ---- ~ = --klnZcEF = - k l n V T - T S ,

(21) E* S* ~CMO = ~* = -klnZcMo = - k ln (T* / V * ) = T*

These equations are appropriate when considering the classical respective ~,canonical ensembles~) of the systems. ZCE F andZcMo are the sum-over-states of our systems. E and E* are their respective thermodynamical energies. S a n d S* are the


thermodynamic entropies. T and T* are the absolute temperatures. V and V* are the CEF and CMO relevant ,(volumes,, respectively. We recall that, since our model is uni-dimensional, their appropriate physical dimension is [lenght]. Thus the corresponding ,(pressures, p and p* we introduce later on are effectively [force]. Idler costants are dropped in eqs. (20),(21). From these equations we also get the state equations:

(22) p V = k T = E ,

(23) p* V * = k T * = E * .

Here p is the CEF pressure and p* the CM0 pressure. So far as regards the unperturbed systems. Now let them interact via

emission/absorption of radiation. As we already noted, with respect to the unperturbed situation characterizing the purely classical systems, this is conceptually accomplished by activating an electrical charge for CMO and injecting a ~,fluctuatiom, energy amount AE into the system.

Let us then consider the following variations of ~CEF, ~CMO:

- p d V d E (24) d~bcEF = k T - E dS = O,

p*dV* dE* dE* (25) d~cMO -- - - - dS* -

k T * E * E *

From these equations forth, dividing by k the thermodynamical entropies is understood. From eq. (24) it is seen that an entropy variation AS taken at constant volume is reversibly accommodated by CEF in a transition between two equilibrium states at energies El, Ef such that

(26)

Ef

f dE = in ~_Af A s = -E-- E~

Now suppose the heat quantity - E dS can (reversibly) be transformed into the work p* dV* during the CEF-CMO interaction. We believe that this assumption can be substantiated by considering the unit value of the black-body absorbance coefficient, and could be a major sophism to discuss this here with reference to Carnot's theorem. However, it will be clear from the following equations (38) and (39) that in our framework a constant Boltzmann energy k T exists, guaranteeing that the overall process can be regarded as an isothermal transformation of CEF + CMO flipping between the extreme states. Therefore, we definitely assume that heat is transformed into work without constraints in the canonical ensemble. From eq. (25) it is seen that the variation of the function h~cMo corresponding to the adiabatical work is reversibly accommodated by CMO as follows:

(27)

E$ Yr

f dE* f dV* E~ , V~ A~CMO= E* = ~ - lnE--~ = m v ~ "

E? V?

1322 6. MASTROCINQUE

We also have from eq. (25):

(28) AS* -- 0.

The CMO entropy variation AS* is taken equal to zero here because the process at hand is ,,work- injection/emission in/from CMO. Complementally, the perturbed potential for CEF is the entropy because the process is ,,heat radiation/absorptiom) in/from CEF. Now we give the Gibbs energies for the extreme (initial and final) out-of-equilibrium thermodynamical states of the overall system (CEF + CMO):

(29) Gi -- E i + Pi V - TSi -t- E * + p~ V~ - TS~ ,

(30) Gf = E f + p f V - TSf + E~' + p~ V* - T S ~ .

In these equations, T is a constant temperature characterizing the process. It is not to be confused with the current values T, T* appearing in eqs. (20) through (25). These last are in turn identical to the thermodynamical energies E , E * themselves. Since only the constant T will be used in the following, no ambiguity can result. We neglected the contribution to the Gibbs energy of the CMO translational degrees of freedom, due to the small momentum transfer involved in radiative processes. From the energy conservation equations (19) and state equations (22), (23) we have furthermore

(31) Pi V + p~ V~' = p f V + p~" V~'.

From the CEF Helmholtz energy equation (20) taken at constant Boltzmann energy and volume we have

Ef- Ei (32) S f - Si = k T

From the analogous equation (21) for CMO we have

(33) S ~ ' - S * = E ~ - E ~ ' , V~ k T m - - ~ = AS* = O.

Using eqs. (29) to (33) we find the following expression for the WJE equilibrium constant:

(34) Kwj E (T) = exp - - ~ = exp [Sf - Si + S~' - S* ] = exp kT "

As is clear from the last equation, we factorized Ke(T) into two terms, i.e.

(35) K e (T) = KCEF KCMO,

(36) [ Ef - E i ]

KCEF (T) = exp [Sf - Si] = exp [ -~- ,

(37) KcMo (T) = exp [$2" - S~ ] = 1.

Although equal to 1, the expression for the CMO-entropy-dependent factor of the equilibrium constant is relevant to our purposes. Now we use eqs. (26), (27), (32), (33) to assess that the entropy variations taken at constant Boltzmann energy have to be


equal to the corresponding ,<continuous path, over equilibrium states variations:

E f (38) AS ---- S f - S i ~-~ E f - E i In , kT E i

(39) AS* = S~' - S* = E* - E~

kT In V2* _- In E__~_~ _ In V~ = 0

V~' E* Y~' '

(40) E~ - E * = In E___~ kT E~" "

Equations (38) and (40) are the thermodynamic examples of a procedure we specifically want to promote in this paper. In this instance the procedure is displayed by the assumption that the variations of the thermodynamic potentials taken at constant Boltzmann energy are equal to the corresponding variations taken along the reversible thermodynamic path between the extreme states. The statement determines here the equilibrium condition in the form of a characteristic energy distribution relevant to the energy values of the extreme thermodynamic states. In the next sections we will conceptually apply the same procedure to the mechanical counterpart of the interacting systems model�9

Now using eqs. (38), (40), (19) we find the energy distributions and the total energy of the system as follows:

(41) E i = E ~ = A E = EBE, exp [AE/kT] - 1

E T = E i + E~ + AE = 2EBE + A E . (42)

In these equations, the subscript BE indicates the Bose-Einstein energy distribution. Now from eqs. (16), (34) we have

[ E f t - E i ] YfN1 (43) K(T) = exp[ ~ =

The last equation can be solved as follows:

1 (44) Ni = Nf = ~ ,

E f

Ei"

(45) N I = I - N 2 = N 2 e x p ( E f - E i ] = k T 1 E l - E i ] '

1 + exp k~'

(46) Ne = 1 - N 1 - -

1 + exp kT

These equations give the thermodynamical-equilibrium values for the number of carriers Ni, Nf, N1, Ne. Equation (42) gives the total (CEF + CMO) system energy. However, we do not have an expression for each of the thermodynamic energies of CEF and CMO, respectively�9 To be consistent in our classical view, we cannot assign

88 - I I N u o v o C imen to B.


to the reagents and products the out-of-equilibrium energies of the extreme states of the fluctuation path except at the start time (see eqs. (17), (18)). After that, indeed, they will undergo changes in their energies as a consequence of the emission/ absorption process. Then each of the carriers will have assigned, in the thermodynamical equilibrium condition, the corresponding mean canonical en2emble energy, which is not given by this thermodynamic model. In order to find these energy values, the following discussion is introduced.

3. - Mechanical -s tat i s t ica l model o f the c lass ical radiat ion process .

In order to display our method, we want to obtain here an expression for the equilibrium constant Ke only using mechanical and statistical (MS) considerations. Although to this end a few known methods of statistical thermodynamics will be used or will appear to underlie our arguments, we purposedly want to let formally apart the thermodynamics in this section. Specifically, we let apart the customary definition of absolute temperature and take different symbols ( E , E * ) for the Boltzmann energies. Then the suitable expression of Ke to our purposes, taken on the basis of the Arrhenius equation, will be written as follows:

m

PfP['QfQ[" Pf(E)Pf(E*)Qf(E)Qf(E*) (47) Ke - =

PbP~QbQ~ Pb(E)Pb(E*)Qb(-E)Qb(-E*)

AE,+ - ] -- _ - PfPf* ~kE+- + _ =KcEF(E,E) 'KcMo(E*,E*). Pb P~' exp ~ E*

As is clear from this equation, Ke is taken here dependent on the current values of its arguments, which can assume both ,,equilibrium, and ,,out-0f-equilibrium~ specifications. In the last case, calling Ke the ,,equilibrium constant- is not appropriate any longer. However, we will do so in the following for the sake of simplicity.

In this eq. (47), we factorize the equilibrium constant into terms dependent on CEF and CMO internal energies E, E* and statistical Boltzmann energies E, E* , respectively. The asterisks are for CMO relevant terms. According to the superscript MS, we want to calculate Pf, P~, Pb, P~, Qf, Q~, Qb, Q~ from mechanical-statistical laws. As already displayed in eq. (47), we take the exponential terms Q's from the classical method of the most probable distribution. This results in the customary fact/interpretation that the probabilities Q to find classical energy carriers able to overcome the activation energies AE • AE *• are proportional to Arrhenius exponentials. Thus we definetely assess to the purposes of this paper the interpretation of the equilibrium constant as the ratio of the forward/backward probabilities for the process to occur. This interpretation turns out to be appropriate--although not necessary--to the application of our following method in sect. 4.

Now we want to calculate the pre-exponential terms Pf, Pb, Pf*, P~ on the basis of a mechanical model and a simple statistical postulate. We calculate the relevant terms to KCEF and KCMO separately, starting with KCEF.

3" 1. Calculation of K C E F. -- Using the forced oscillator equation for a linear Hertz dipole interacting with an isotropic incoherent radiation field Planck found a


celebrated expression for the oscillator energy Up:

C 3 (48) up =

87ZV 2 U~

In this equation, c is the light velocity and uv is the field energy density per unit volume and frequency. Equation (48) is calculated assuming a stationary forced solution, so that u~ is taken unperturbed by the interaction itself. Noteworthy, Planck's universal equation takes into account the (small) radiation reaction term. The same result (48) is also found independent of the initial oscillator state. In this paper, we will remove the approximation that the field energy is unperturbed by means of our fluctuation model. We will aiso introduce a specific point about the internal oscillator states relevant to the calculation of KcMo. This will be done by means of a simple mechanical-statistical model introduced in the next section.

Since eq. (48) is independent of the initial oscillator state, it can be interpreted in the sense that an incident field on the Hertz dipole mechanically transfers to the latter, in the unperturbed approximation for u , the energy Up. Straightforward application of this result to our uni-dimensional case can be expressed as follows:

(49) hEc = aE, a = const.

hEc is the energy made available to CMO via the classical energy transfer equation when CEF is switched on.

Let us now introduce the following classical probabilistic postulate in order to be consistent with a statistical theory of energy transfers.

Given the energy amount AE c classically transferred during the interaction from CEF to CMO, the probability that an otherwise specified energy amount AE is transferred is postulated as proportional to this mechanically available energy hEc:

(50) Pb(E) = bAEr = L E h E

In this equation, L is a suitable coefficient to normalize the probability function. It will be determined by the following procedures in the paper.

Given the backward process (here the absorption) probability in eq. (50), we can also write the forward process probability as follows:

(51) P f ( E ) = 1 - P b ( E ) = 1 - L__EE hE,"

In this equation, we used the fact that the probabilities that an energy amount AE is emitted (Pf) or absorbed (Pb) by CMO are clearly ,~mutually exclusive, (the energy AE is either in CEF or in CMO).

From eqs. (47), (50) and (51) we obtain the following expression for the equilibrium constant:

L E

(52) KCEF (E, E) = (E----~ ~ -

AE


The Arrhenius exponential term in eq. (52) takes into account the energetic disadvantage affecting the carriers involved with the absorption process with respect to the emission.

3"2. Calculation of KcMo. - Let us now proceed to the calculation of KCMO (E* , E* ). Since general information for anharmonie oscillators is not available, we will strictly specify the calculation for the harmonic oscillator case (CHO). We have from the classical theory of radiation by an accelerated charge (spontaneous emission, SPE):

E* At (53) AEc - T c

In this equation, ~r is the classical oscillator lifetime, At is a small integration time and E* is the classical, unperturbed CHO energy. We note that the classical equation (53) can be integrated in the time domain, but according to the standard perturbation theory we take here E* as a constant and apply the probabilistic postulate:

AEr L ' E * (54) PSPE (E* ) . . . .

AE AE

Equation (54) is analogous to eq. (50) and means that the (classical) SPE probability to emit the energy amount AE is proportional to the available internal energy E*. The coefficient L* does not need to be determined in the sequel.

Let us now introduce the following mechanical-statistical model for the classical harmonic oscillator interacting with CEF. Suppose the mechanical energy level E* of CHO is twice degenerated, according to the fact that the incident CEF is in phase or is dephased by = with respect to the oscillatory motion. Then the simplest MS model of CHO is based on the calculation of the sum-over-states as follows:

(55) Z* = 2 exp -

m

In eq. (55), according to our proposals, we used E* for the Boltzmann energy, for later identification with kT. From the same equation we find

(56) ~* - k In Z* E* E* . . . . k l n 2 = _ - - - - - S * . E* E*

So we identify the statistical mean energy of CHO as E* and we find the corresponding entropy S* = k ln2 . Let us now introduce the CEF as a perturbative agent. According to the fact that CEF is able to force CHO into two different states by doing mechanical work (positive or negative) on it, the energy level E* will split into two energy levels as follows:

(57) E* = ( E * ) _+ W.

In this equation, W is the (absolute value) mechanical work made by CEF. This quantity is not determined yet, and we will take the energy E* -as a fluctuating


quantity. Then the sum-over-states can be written as follows:

(58) Z*' = exp - + exp ~ , .

Now we consider the following. When stimulated emission (forward process) is likely to occur, the CHO energy is ,(high~, the plus sign applies in eq. (57) and the spontaneous emission probability is

(59) p~ L*(2(E*)-E*) L* ( L**E* ) SPE---- hE -- L** I hE ,

(60) L** = hE 2(E* )"

The inverse is true for the backward process (absorption): the CHO energy is ,,low~, the minus sign holds in eq. (57) and the SPE probability is

(61) P~PE = L'E* hE

Our main postulate now is expressed by the belief that eq. (58) is perturbated by the SPE effect because this introduces the following ,,mechanicab~ entropies to be taken into account when writing the sum-over-states:

L* ( L * * E * (62) S~pE = In P~PE = l n ~ u + In 1

(63) S~rE = lnPsbPE = In L* E_____~* hE

Taking these equations into account, the sum-over-states becomes

(64) Z* ' --~ Z*" = exp ~ , + S - + exp ~ , + S + =

___L'E* [ E~** exp - + L*(2(E* ) -E*) [ 2(E* ) - E * exp ~ , .

By inspection of this equation, we see that the ,ca priori)~ probabilities Pf(E*), Pb(E*) of finding CM0 in the appropriate state to react (,forward~ or ,,backward~ with CEF are

L* ( L**E* ) (65) Pf(E* ) = ~ 1 AE '

(66) Pb (E*) = - - L'E*

AE

The Arrhenius Q* needed in eq. (47) are also available in eq. (64). Using eqs. (64)-(66)


in the KCM 0 expression (47) we finally obtain

- - L * ( 2 ( E * } - E * ) [ 2 ( E * } - 2 E * ] (67) KCM 0 (E*, E* ) = L* E* exp - ~ . =

L**E* 1 AE [ ( 2 ( E * } - 2 E * ) ]

- L**E* exp - ~ , �9

AE

Now we write here the final expression for Ke taking into account eqs. (52) and (67):

(68) K e ( E , E , E * , E * ) = KCEF(E,E)KcMo(E* , E * ) =

LE L**E*

_ _ _ 1 AE exp L**E* exp _ . LE E* hE

This expression for the equilibrium constant is dependent on the unknown coefficients L,L** as well as on the relevant CEF and CM0 energies. In order to determine the mean values of E, E, E * , E* to be inserted in the equilibrium constant expression and the values of L,L** the following discussion is introduced.

4. - T h e o r y .

According to our fluctuation model of inelastic processes, the energy transfer probabilities (and consequently, the MS equilibrium constants) customary calculated in the assumption of unperturbed states and constant energies have to be corrected due to inelasticity. The symmetrization rule in eq. (4) is appropriately applied to both KCE F and KCM 0. Let us consider first the expression of KCE F (E, E). We note that both the arguments of this function, i.e. the mechanical CEF energy E and the statistical Boltzmann energy E fluctuate between the extreme values Ei, Ef.

Applying eq. (4) we get

(69)

f

KS,CEF (E, El, Ef) = exp In KCEF (E, E) dE

Ef Pf(E) Ef Ei Pb(E) Ei

(70)

LE 1 - - - -

hE LE '

hE


In eqs. (69) and (70) the S subscript indicates symmetrized quantities. In eq. (69) we symmetrized the equilibrium constant with respect to the fluctuation of the Boltzmann energy E and in eq. (70) we symmetrized the resulting expression with respect to the fluctuation of the total energy E. The resulting expression for KS,CEF is seen to depend symmetrically on the initial and final energy values Ei and El, thus respecting the detailed balance principle. The function S introduced in eq. (70) results from integration of ln Pf/Pb as given by eq. (52) and is found to be

(71) S(P) = -P lnP - (1 - P) In (1 - P) .

The function S(P) is recognized as the Boltzmann statistical entropy for a binary process occurring with probability P.

According to eqs. (4), (5) and (6) we now define the ,collisional, or (,mechanical. entropy (difference) A S c ( E i , E f ) for the process at hand as follows:

(72)

l LEf - S

AS c (El, El) = lnKs,cE F (Ei, El) = L

Ef + In E--~."

We will use these equations later on. Let us now apply the symmetrization_ rule (4) to the quantity Kc~o (E*, E* ) expressed by eq. (67). Again, both E * and E * are seen to fluctuate between the extreme values E~*, E~'. We have

(73)

Er KS'CM~ ) = exp[ I ! (E*, -E* ) d~, *] =

1 - L * * E * hE

L**E* hE

[ E2 ]-(2{E*)-2E*)/~E

E ~ J

In the previous equations, we used our symmetrization rule with respect to the variables E, E, E* taken as independent of each other. As far as the quantity E* is concerned, we note that it is dependent on E via the total energy equation (19). Thus we take dE* = - d E and the Kc~o symmetrization results as follows:

(74) Ks,cMo(E*,E~)= e x p ~ - ~ [lnKs,cMo(E*,E~,E~)dE* = k E~

= exp -

L** ] [ E~'

1330 (;. MASTROCINQUE

In this equation, similar to eq. (70), the function

(75) S t - S - AE , AE

L**E2 [ L ** E* in L**E* t AE AE ~ AE

ln(1 L ** E,2AE ) +

+ - - L**E~AE .ln--L**E~AE +(1 L**E~) ( _ _ A E In 1 L**E*)AE,

is recognized to be the difference in Boltzmann entropies for binary processes occurring with probabilities L** E2/AE, L** E*/AE.

This leads to the interpretation that the <<mechanical,> disorder corresponding to the internal CM0 energy fluctuations appears here as the Boltzmann statistical entropy difference correlated to the spontaneous emission by the extreme fluctuation states. In an analogous manner to eq. (72), now we define

(76) ~ * (E*, E~') = In gs,c~o(E~', E2*).

This equation will be used later on. We are going now to introduce our main statement in the paper, which we propose

to consider as an interesting property or perhaps the principle of the thermodynamical equilibrium of our systems as follows.

We say that, at the thermodynamical equilibrium, the ,collisional- entropies ASr AS*(E~,E~) introduced by the energy transfer process, which we calculated by means of a mechanical-statistical model, have to be equal to the thermodynamical entropy variations for CEF, CM0 respectively, displayed in eqs. (32), (33). We could also say--perhaps more simply--that the corresponding MS equilibrium constants in eqs. (70) and (74) have to be equal to their thermodynamical counterparts displayed in eqs. (36) and (37).

Although gcE F and KcMo are not independent of each other because of eq. (19), it turns out that we can separately consider the CEF-and CM0-dependent parts of the entropies or equilibrium constants. We consider first the expressions (36) and (70) for CEF. We obtain

(77) KCEF(T)=exp[Sf-Si]=exp[~]=Ks,cEF(Ei,Ef),

(78) E f - E i ] Ef

exp ~-~ = E i i exp L

This last equation can be solved as follows:

(79) Ei exp - ~

A PROPOSED PRINCIPLE OF THERMODYNAMICAL EQUILIBRIUM ETC.

L

1331

From these equations, we find

(81) E i --"

(82)

hE exp [AE/kT] - 1

= E B E ,

L = AE/ (Ei + E~) .

The result in eqs. (81) is confirmed by previous considerations in this paper. So we are encouraged to apply more extensively our procedure.

Let us consider the CM0-dependent part of the equilibrium constant. Applying again our main statement in the paper, we say that the detailed-balance

value of Ks,c~o in eq. (74) has to be equal to the thermodynamical counterpart KcMo (T) given by eq. (37):

(83) Ks,cMO = KcMo (T) = 1.

This equation is easily found to be solved taking in eq. (74):

hE hE (84) L** = - - - 2 ( E * ) E ~ + E ~ '

(85) ( E * ) = E* + E~' _ Ei~E + AE 2 -2-"

Equation (85) is the most relevant here, since it gives the CMO mean canonical ensemble thermodynamical energy. This is found as a Bose-Einstein term plus a zero-point energy. Now we can find the mean CEF energy from the total energy equation (19), but first we consider the following. The (,uncorrected)) value of KCE F (E, E ) is seen to come useful again if we say that, when this quantity is taken as a function of ,(meam~ quantities (E) , (E) , its value is expected to be equal to the ((true~) value KS,CEF (Ei, E~). Thus we write

(86) KCEF ( (E) , (-E)) = KS,CEF(E~,Ef) = KCEF (T) =

1 - L (E)

L (E~ exp = Eli exp

hE L

E f

E i "

We can solve as follows:

(87) (-E) = A E / l n E f / E i = k T ,

(88) ( E ) = E f h E = E i + ' ~ ' 2 T = EBE + - -

AE 2 "


We also obtain

(89) 1 (Pf) = Pf((E}) 2 '

(90) 1 (Pb) = Pb((E}) = ~ .

Equation (88) is the most important here to our purposes. It gives the expression of the mean canonical ensemble energy of CEF, consistent with eqs. (19), (85). This energy is found out as a Bose-Einstein term plus a zero-point energy AE/2. We note that taking in this equation hE = ho~, with ~ the oscillation frequency of CEF and CM0, immediately leads to the Planck distribution for the single-mode CEF energy in equilibrium with the black body, inclusive of a zero-point energy. Major interpretation of this result will be given later. Now we note, by using the ,~unsymmetrize&~ expression for KcMo ( ( E * ) , ( E * ) ) taken as a function of the mean values of its arguments, that we find

(91) KcMo ( (E* }, ( E * ) ) = Ks.cMo(E ~ ,E~) = Kcmo (T) = KS,CMO ( ( E * ) , E ~ ~ , E ~ ) = 1.

From this equation and eqs. (66), (74) we have

(92) ( E * ) = a~,/m~--~- = kT,

(93) (E* } = E* +E~2 =E~E+ ~hE

These last equations merely give in this instance the information we already obtained in eqs. (40), (85). However, here it is independently obtained. This fact confuzns some outlines of our procedure.

Taking all the results of this section into account, we recognize now that the out-of-equilibrium rate equation for the balancing of hE between CEF and CMO can be expressed as follows:

(94) E ---- KfNiN 2 -- gbNfN1,

(95) E (1 E* ) [ E * + E ~ - 2 E * J Kf = 8B exp [AE/~] ETT E* + E~' s i exp - ~2~* '

ET- E E* E i [ E~ + E~ - 2E* ] (96) Kb = 8B E ~ E~' + E~' exp 2E* "

In these equations, B is the characteristic process rate, easily seen to coincide with the Einstein stimulated emission coefficient. Taking indeed the equilibrium values for E, E * , E, E * , Ni, Nf as given by the respective eqs. (88), (93), (87), (92),


(44) we can approximate eqs. (94)-(96) as follows:

(97)

" E i + E~ E = ( E } = 2 '

E f exp [AE/(E}] = -~i'

(Kf } = 2BEf, ,

E'* = {E*) - E ~ + E * 2

exp [AE/(E* }] = E2* E ~ '

1 (K b } = 2BEi, (N i } = (Nf } -- ~ ,

(98) E = d(E} =L, i=B(E~N2_EiN1)=BEi (N2_N1)+BAEN2 = _AE_N2 dt

Equation (98) is found identical to the Einstein equation for emission/absorption of radiation, where the quantity Ei is generally interpreted as the CEF energy straightforwardly. In the present frame, it is instead the low extreme energy value of CEF fluctuations. We recognize that in our model eq. (98) also provides the appropriate rate equation to .match the Boltzmann energies (E}, (E* } with the thermodynamic equilibrium value kT. Equations (19) and (97) can be used to display the relevant dependence of these quantities on the variable El . We also note here that the last term in eq. (98) is obtained by considering that varying the number of emitters implies proper accounting of the CEF energy. The solution of equation (98) is a standard matter but for the sake of completeness we stress again the following properties holding at the thermodynamical equilibrium:

(99) t7 = E i = o = = k T ,

(100) Nz = exp - = exp - . N1 kT

We have to note here that eq. (100) is interpreted as due to the Helmholtz energy difference characterizing the extreme fluctuation states by which the emitters and absorbers are distinguishable in our frame. Both these classes of CMOs have indeed the same thermodynamical energy given by eq. (93). This is because the equilibrium condition is a ,,dynamicab~ one, i.e. a situation where the ,~emitter, fluctuates from the high energy E* to the low energy E* and then back to E~', and the ~,absorber,, vice versa, starting from E~. Now we note that when the stationary temperature state in eq. (99) is achieved by the system via the Einstein equation, still we can use eqs. (94)-(96) to describe the out-of-equilibrium internal energy fluctuations of, e.g., CEF as follows:

(101) ~'[r=const = 8BE~N2 1 E~ + E~

E ~ + E ~ - 2 E * 1 E T - E E* �9 exp - 2kT 2 E T E* + E~ Nf (t).


[ E* + E ~ ' - 2E* ]} 4B(ET - EBE) f ( E , T , t ) = B f

�9 exp 2kT E~ 1 + exp

(102) f (E, T, t) = (t)E ~ exp [ ~ - ~ - Nf(t)(ET - E) 2 exp ET -- 2E i

In eqs. (101) and (102) we displayed a time dependence of Ni, Nf although by their thermodynamical properties they are essentially constants. This time dependence has to be understood as a ,,square-wave~ function, flipping between the values (0,1) and changing state each time that the fluctuating energy E reaches the extreme values Ei, Ef. The two functions for Ni, Nf assume complementary values so that either Ni is 1 and Nf is 0 or vice versa. It looks consistent indeed in our frame to take the CEF carriers reacting with, e.g., the absorbers as living the time during the transition from Ef to Ei, and complementary. Then from eqs. (101), (102) one finds the fluctuation frequency as follows:

B exp [ET/2kT] 2BkTE~E (EBE + hE) (103) ~if(T) = E~ hE(2EBE + hE)8

2s(T) f exp [+E/kT] E2 dE

Ei

where tiT) has been introduced in eq. (101). The frequency v~ is easily seen to be proportional to T at high temperature and to go rapidly to 0 in the low-temperature limit.

Now we conclude this section by considering the following. Concerning eqs. (88), (93) we note that, in modern mechanics, in order to introduce a zero-point energy in the physical systems, one is obliged to use ,(quantizatiom> procedures. The zero-point energy we obtain here, however, is not identical to that relevant to the Heisenberg principle. No constraint has been imposed indeed on the value of AE here. Although with our procedures we practically recover the Einstein equation, consistent with the quantum point of view, we recall that the model here introduced is the classical model of the interaction between CEF and CMO. We only ask to CEF and CM0 to be ~,resonant~. When we take hE = ho-- that is a more specific point--then we start with quantum mechanics. For in the latter case we recognize that this zero-point energy still exists for the two systems separately, i.e. without imposing them to interact. Precisely this point leads us, at the end of this analysis, to the conclusion that a physical model for mechanical oscillators may exist where the ,dsolated~ classical oscillator interacts with what we could call a ,~fluctuation field~ able to exchange the energy amount h~ with CMO, approximately in the same way as we showed here for CEF. The classical, mechanical laws of the energy transfer could then be used again to describe, via the ,(ergodic)) postulate, this energy exchange. This model would lead to the complementary insight into general physics with respect to wave mechanics: the ~,discrete~ or ~quantum~ spectra of mechanical oscillators would arise as the statistical mean values of an energy levels ,~ensemble~ fluctuating around the corresponding (,eigenvalue~. We give in the next section the simplest of the mechanical oscillator models based on this view.

A PROPOSED PRINCIPLE OF THERMODYNAMICAL EQUILIBRIUM ETC.

5. -- The mechanical oscillator model.

1335

Let us consider a classical, unidimensional oscillator with mass m, potential function U(x), total energy E and oscillation period T(E) given by

x2 (E)

(104) T(E) = ~ f dx ~1(~) ~/E- U(x)

In this equation, xl (E) and x2 (E) are the energy-dependent turning points. Note that we will omit the asterisks for CM0 in this section and in sect. 6. We suppose that the mechanical energy of CM0 is not a constant, but fluctuates by an amount AE, continuously assuming all the values in the interval (El, El). This energy amount is transferred reversibly, in our model, to a ,,fluctuation field- (FF) able to store it. We suppose that mechanical laws exist--not known yet--governing the energy exchange. According to classical mechanics, the probability Pc (E) that the exchange occurs can be written as follows:

1 (105) Pc (E) - 2MAE I S F(t) exp[iot]dt .

- - a o

In this equation, o and M are a frequency and a mass to be attributed to the FF, and F(t) is the time-dependent force exercized by the oscillator on F F (or vice versa). Although this function F(t) is unknown, we can estimate, according to standard methods, the exponential dependence of Pc (E). Since indeed F(t) depends on CMO oscillation dynamics, we can write

(106) [ �9 2=t ] F(t) = A(t)exp[-~ T-- ~ ],

In eq. (107) we evaluate the force integral according to standard saddle point approximation (SPA). This features the integral as an exponential form, where z is a characteristic interaction time. The function A(t) and the mean value A are unknown. However, taking the characteristic time z equal to the CM0 period T(E) leads to the following expression:

(108) Pc (E) ~ exp [2[~oT(E) - 2=]].

Here we completely neglect the pre-exponential part. Since E is taken constant as if CMO is unperturbed by the interaction, we symmetrize eq. (108) according to our


fluctuation model. We have

(109) [ i f 2~o T(E) Pc(E) =>Pif(Ei,Ef) -- exp -~E~ dE - 4=].

We also define the interactive entropy as

(110)

Ef

2 ~ I T(E) dE - ASc (Ei, Ef) = lnP~ = ~-~ 4=

Ei

As already shown in the previous sections for the case when CM0 interacts with CEF, we can equally assume here that the mechanical energy fluctuation is due to work injection by FF to the oscillator. Thus the thermodynamical entropy ST of CMO is assumed unchanged. According to our proposed principle, then we obtain that the interactive entropy ASs, on the basis of the equivalence with the thermodynamical correspondent AST, has to be zero:

(111) A S c ( E i , Ef) = ~ T ---- 0.

From this equation

(112)

Er

f T(E) dE = 2=~E _ h.

Ei

In this equation, we definetely display the assumption that the FF carrying the energy AE has the frequency ~ / h . Then we recognize in the last equation the old, celebrated Bohr-Sommerfeld rule. This is a sort of ,,mean resonance condition-, which we can interpret in the present frame as an ,,adiabatical- condition (AST = = 0).

Now we still can use, as already shown useful, the nonsymmetrized entropy or fluctuation probability to find the mean value of the system energy during the fluctuation. We write

(113) ASc((E)) = lnPc((E}) = ASc(Ei,Ef) = 0,

(114~ lnPc ((E}) = 2(coT((E)) -2=) = 0,

(115) T ( ( E ) ) - 2= _ h ~o AE

Except in the case where the oscillation period does not depend on the energy, this is an useful equation. It is recognized to give the oscillator quantum-mechanical energy eigenvalues, given the energy spread AE from the Bohr-Sommerfeld rule, in the quasi-classical approximation.

However, we are specifically interested in the CHO case, where, due to degeneration, eq. (115) is useless. On the basis of the results we obtained in the previous sections of this paper, it is easy to extend the definition of interactive


entropy hSc in this case as follows�9 First of all, we consider the following quantity:

~ f ~ d ~ / E - U(x) _ ~ dE 1 (116) dp(E, x) = T(E) ~v/E - U(x) T(E) "

This quantity is the variation of the oscillator momentum mv(x) taken with respect to the total energy, and divided by the oscillator period. It can be interpreted as the effective resistent force locally developed against the changes of energy from E to E + dE. Then the quantity

(117) f dp(E,x)dx = f dE dx d_..__~_t _ d E v(x) dt T(E)

E E

is the total work necessary to change the energy by theamount dE during a period. Since it is equal to dE, we conclude that the period is the characteristic time for the mechanical oscillators to change adiabatically their energy. Let us take this way: when dE is negative, the energy is emitted and is collected by FF in our model. We do not know the mechanical emission law but note that for the degenerate case of CH0, since the period T is independent of the energy, same amounts of energy can be emitted in the same time interval, independently of the initial available energy. Then in the harmonic-oscillator case the mechanical ,,emission, probability can be straightforwardly taken proportional to the available energy as follows:

(118) Pf(E) = L__EE hE

For the reverse process, absorption from FF, we will write

(119) PD(E) = 1 - P f = 1 - L E hE

Following procedures now identical to those we illustrated previously, we can assess the expression of the equilibrium constant for fluctuations between CMO and FF:

(120) Ke -

L E

LE exp 1 -

AE

T(E)dE - 47: .

k ~i J

In this equation, although we referred to the CHO case, we maintained for the sake of generality the eventual energy dependence of the oscillation period. In the same equation, it is recognized that the pre-exponential part is assumed on behalf of the unknown quantity ~2 in eq. (107). F F energy parameters do not appear in eq. (120) due to incomplete information. However, if we assume eq. (120) as exhaustive, we can find the CHO energy if we impose the symmetrization rule (4) and the adiabatical


condition (113) by taking into account the pre-exponential part of Ke. We find

(121) + hE2~~ ~__ T ( E ) e l N - 47: ,

Ei

where S is identical to the Boltzmann entropy defined in eq. (71). Using now eq. (113) we find

(122) lngs(Ei , El) = ~S e = 0 ~ Ks(El, Ef) = 1.

Using eq. (112) we have

(123)

From eq. (123)

hE Ei + E f "

(124) L =

Using now again the procedure for the mean values, arguments of the unsymmetrized Ke (( E ))

(E)

Ei + Ef (125) g e ( ( E ) ) = 1 =

(E) ' 1

E i + Ef

(126) (E) = 2 = - ~ + hE,

Finally, we can again assume equation (40) because this holds independently of FF thermodynamical behaviour and we obtain

(127) (E) = EBE (T) + ~ 2

This is the mean CM0 thermodynamical energy. It is found out to be identical to the thermodynamical energy of the mechanical Planck oscillator, plus the zero-point energy, if we take hE = ho~.

6. - - T h e t h e r m o d y n a m i c a l o s c i l l a t o r m o d e l

On the basis of the previous results, we conclude that entropy is correlated not only to the classical concepts of ~,disorder>,, <,fluctuations,>, or ,<log-number of available states~, but also to the concept of <(energy transfer probabilities,,. As a consequence, we are encouraged to define here more completely the thermodynamic model of CH0 resulting from this paper and accounting for this interpretation. Our CHO is different from the purely classical harmonic oscillator because it is able to


exchange energy with a ,,hiddem) degree of freedom called FF. This property is accounted for by an effective (~volume, term displayed in its thermodynamic function, the same that we used in sect 2.3 to allow interactions with CEF. We can call this oscillator BHO, ,(the Bernoulli oscillator,. We resume here a few thermodynamic properties of BHO:

T - S = I - S . (128) ~bBHO = --lnZBHo = --ln ~ = - l n P = - l n E E V T

Here E is the thermodynamic energy and P is the pressure. The Boltzmann constant and idler constants are always dropped. We want to show now that BM0 can assume the thermodynamical properties required by quantum theorists if operated as follows.

Since the entropy S is correlated to the probability of an energy transfer from BM0 to FF (and vice versa) we give the interpretation of eq. (128) as an ,,unperturbed~ result. We can take into account the energy fluctuations by variating eq. (128) in an identical manner as we did in eqs. (25), (27), (33), (39), (40) and by ~symmetrizing, the function e s. The quantity AE can be taken from the Bohr-Sommerfeld rule in eq. (112). We obtain

(129) E i = E f - h E = h E = E B E ,

Ef Ef

if (130) ( S } = - - S d E = l + i n = : d E = hE

Ei Ei

E f E i - - - l n E f -

hE hE lnE i - lnV = S(V , E i , E f ) .

Now we replace with the ((symmetrized)) S the entropy in eq. (128). We have

(E} (S). (131) --lnZBHo--> -- lnZslBHO = T

By standard thermodynamics, now we find (E}:

,. d(E} d(S~ dEi (132) (E) = T 2 - ~ lnZs, BH0 = (E) - ~ + T 2 dE i d T '

(133) d(E____~) = dE___A dT dT '

(134) (E) - Ei + Ef 2

+ const = EBE + const.

Integrating (132) with (134) we could find an explicit expression for ZSIBH 0 but first we consider the following. The value of the constant appearing in eq. (134) is not determined by the procedure but we recall that ref. [1] gives a more specific (although

89 - I l Nuovo Cimento B.


less rigorously determined) symmetrization procedure involving the case of a ,,slow,, variation or pre-exponential part of the fluctuation probability. We find indeed that exp [ - ( E ) / T ] has a fast variation with respect to Z. Thus we have [1] that ZBH0 is appropriately symmetrized by taking the arithmetic mean of the extreme out-of-equilibrium values Zi,Zf. We find, using the ,,second), symmetrization procedure,

(135) 1 Ei 1 Ef

(ZBHo > = ZS, BHO = ~ In V + 2 In -~-,

Ef

l i e , , Ef (136) ( S > = - ~ j 1 - - + - 2 1 n V + 2 1 n V = Ei

E~ + E~ + In " - ' - ' - S(V,E~,Ef).

2T V

The last term in eq. (136) is implied because expression (136) is found identical to eq. (130), guaranteeing constistency to the overall framework. We note here, as a reference for further investigation, that the first procedure leading to eq. (130) is strictly developed using an ensemble of equilibrium states, while the second one, leading to eq. (136), is developed through a path involving constant parameters pertaining to the out-of-equilibrium states.

Now, since we find that the term In V is redundant in eq. (136) we definitely recognize that the thermodynamical function of BH0, i.e., the ,,classical, fluctuating~ harmonic oscillator is

In EVEiE~_ hE 1 (137) ~bSlB~O = --lnZs, BHO = hE 2T In 1 - exp [-AE/T]

_ EB____EE + /kE Ef In Ef + E i In Ei + In AE. T 2T hE hE

In this equation a term In AE has been inserted to take the Third Law into account. By comparison, we find that const = 0 in eq. (134). So our final result is that eqs. (134) and (137) are identical to those pertaining to the quantum harmonic oscillator if we drop idler constants and take AE = ho~. This result looks as the most important in the paper, since it is obtained using ~classicab~ arguments.

The procedures and ideas we propose in this paper are possibly sometimes ,,not orthodox,~. Certainly, some details Or arguments can be changed or improved. However, the overall framework seems suggestive enough to require further investigation. We can conclude that our method looks consistent, and give the final interpretation of our equations as follows. In our model, the classical mechanics is able to take into account inelastic processes and energy fluctuations. The classical point of view is then brought to consider the mechanical oscillators as continuously exchanging energy with a ,,fluctuation field~). The mechanical and thermodynamical

A PROPOSED PRINCIPLE OF TttERMODYNAMICAL EQUILIBRIUM ETC. 1341

energy eigenvalues are the mean energy values assumed during the fluctuation. The fluctuation energies AE are found on the basis of the ,,adiabaticab> condition, because the FF only takes/injects work from/into CMO. Although our equations are limited by assessed approximations, we think that improvements can be achieved in our physical model as well as in practical calculations to the purpose of approaching the exact quantum-mechanical results for anharmonic oscillators by means of a dynamical interpretation of the mechanical equilibrium laws.

Accomplishing this task might give new light on quantum mechanics itself, based on a competitive model obtained by means of classical physics.

7. - C o n c l u s i o n s .

We proposed in this paper a ,,classical,> alternative to Planck's procedure. The Planck procedures definitely led to the discovery that a discrete energy spectrum is the theoretical basis to explain the experimental black-body behaviour. Our procedure leads to the alternative interpretation that this discrete spectrum is given by the mean values of the mechanical energy fluctuating over continuum energy intervals, thus providing a dynamical interpretation of the microscopic mechanical laws. These two points of view are complementary. We showed the fruitfulness of a method consisting in calculating detailed-balance equations for the energy transfers, and equating to the unbalanced equations when taken as functions of mean arguments. This point of view seems to be excellent when we take these arguments from equilibrium thermodynamics, so that we proposed to consider the procedure as displaying an interesting, general principle. We gave a few important examples of the utility of this view. The method is also appropriately resumed saying that the perturbation theory is operated with respect to the mechanical laws of the energy transfer in the same way as fluctuation theory does with respect to the equilibrium thermodynamics: the prescription is just to calculate variations of different quantities letting unperturbed something else. This leads the system <<out of equilibrium>>. Then we have to recover the equilibrium state by accommodating the variation in the spectrum of the possible equilibrium states. This is found possible if the ,<balanced>> rate-constants of the process can be put equal to the <,unbalanced>> ones. On this basis we displayed our equations. The radiation problem by a harmonic oscillator looks solved by adopting this view. Simple models for the mechanical and thermodynamical harmonic oscillators are proposed, following analogous methods. Although limited by assessed approximations, they give the correct results. The mechanical model for an anharmonic oscillator only takes advantage here of approximated equations, leading to the known results of the (,quasi-classical case,>, although interpreted in a new light. We believe that extension of our ideas and further investigation might lead to more important results.

We are pleased to acknowledge useful comments by Prof. L. Galgani.

1342

R E F E R E N C E S

G. MASTROCINQUE

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69. [4] W. NERNST: Verh. Dtsch. Phys. Ges., 4, 86 (1916). [5] L. GALGANI: Nuovo Cimento B, 62, 306 (1981). [6] L. GALG~NI: Ann. Found. L. de Broglie, 8, 19 (1983). [7] T. S. KUttN: Black-Body Theory and the Quantum Discontinuity, 1894-1912 (The Chicago

University Press, Chicago, 1987).

a proposed principle of thermodynamical equilibrium and the classical theory of the black body

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