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Mesoscopy and Thermodynamics EPJ manuscript No. (will be inserted by the editor) Mesoscopy and thermodynamics L. G. Moretto, J. B. Elliott, and L. Phair Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Received: date / Revised version: date Abstract. The interplay of thermodynamics and statistical mechanics is discussed with special attention to mesoscopic systems and phase transitions. PACS. 25.70.Pq – 05.70.Jk – 24.60.Ky – 64.60.Ak 1 Introduction course of theoretical speculation is wisely trimmed by the correctness of thermodynamical empiricism. Thermodynamics is an empirical science devoted to the The “recent” extension of experimental [1–3] and the- description of macroscopic systems. It follows formally oretical research to systems that are neither wholly mi- from its three principles and stands firmly independent croscopic nor macroscopic, thus aptly named mesoscopic, of any microscopic description. Statistical mechanics, on should be enlightened by the same wisdom. While names the other hand, is founded on the general principles of and functions (temperature, entropy, free energy...) were mechanics, and attempts to bridge the gap from the mi- borrowed from thermodynamics, the urge to label the field croscopic description of a system to its macroscopic ther- as “novel” led to rash statements such as the primacy of modynamical properties. This attempt has met with un- one ensemble (microcanonical) over the others, or even deniable success through the introduction of ensembles to the affirmation of a “new” thermodynamics or to the such as microcanonical, canonical, grand canonical, etc., “failure” of the “old” thermodynamics and its principles. whereby microstates, constrained to satisfy some specific These rash statements have often been accompanied macroscopic conditions are used to obtain average values by a cavalier disregard for the physical definitions and of macroscopic variables. Its success, however, must be identification of phases, claiming the presence of phase measured in terms of its agreement with thermodynam- transitions on the basis of possibly myopic interpretations ics, which is the final judge. Thus the potentially unbridled of anecdotal numerical results. 2 L. G. Moretto et al.: Mesoscopy and thermodynamics The resulting rush to calculate properties of a plethora any given energy. The result of these calculations is the of individual mesoscopic systems underscores the essential level density ρ(E) as a function of energy, or, equivalently, limitations of these efforts, which renounce the beautiful the entropy S(E) as a function of energy. Anomalies ap- generality of thermodynamics for results as limited and pear in the dependence of S(E) on E, like regions of con- specific as the systems under consideration become legion. stant slope, negative first derivatives and positive second This direction leads to as many individual “statistical me- derivatives (concex intruders). The elementary nature of chanics” as there are mesoscopic systems. We shall discuss this procedure seems to ward off any confusion. this point further and suggest a natural solution. Yet the transition from microcanonical to the more The study of numerical rather than analytical methods commonly used canonical language and variables becomes has given rise to two different problems. On one hand, the dangerous and misleading in correspondence to the very size of the system is finite, on the other, the choice of the anomalies for which one is searching. Thus the resulting algorithm becomes of paramount significance. negative temperature, negative specific heat, etc. tend to create misunderstanding and confusion. The first problem, which leads to the need of extrap- Our own approach uses the reductionistic device of de- olating correctly to the thermodynamic limit, is a nega- scribing the plethora of mesoscopy in terms of the “old” tive aspect of these numerical computations. On the other thermodynamics, which is augmented by the least num- hand, the ever increasing importance of microscopic sys- ber of new quantities necessary to describe the finiteness tems in experimental and applied science, makes the finite of the systems. This idea, of course, is not new. It is em- size of great interest in itself, and the attending differences bodied, for instance, in the “liquid drop” model, so much from the thermodynamic limit are drawing ever increasing successful in nuclear physics. It consists of expanding ther- attention. The second problem is related to the first. Some modynamic functions such as the free energy in terms of algorithms strive to compute the partition function versus a “finiteness” variable A−1/3 where A is the “size - mass” temperature (in the canonical approach), while others try of the finite system. So we write: to compute the entropy versus energy (the microcanoni- 2/3 1/3 cal approach). A lot of discussion has arisen concerning F (A, T ) = av(T )A + as(T )A + acA + ··· (1) the merits of the two approaches, in particular with re- where av(T ), as(T ), ac(T ) are the temperature T depen- gards to the observation of interesting anomalies and to dent volume, surface, curvature, etc. free energy coeffi- the extrapolation to the thermodynamic limit. cients. The leading term is the extensive volume term In the struggle between the two camps, the upper hand av(T )A, the others (which can be shape dependent, e.g. 0 seems to be taken by the microcanonical algorithms which as(T ) = as(T )f(s/s0) where s is the actual surface s of just generate and count configurations in phase space at the object and s0 is the surface of its equivalent sphere) L. G. Moretto et al.: Mesoscopy and thermodynamics 3 are the terms that account for finiteness and vanish for The lack of particle or energy exchange with the environ- A → ∞. The shape variables can either be fixed, or let ment makes the calculational aspect particularly transpar- free to assume the value demanded by the free energy ent. Literally one counts the number of configurations of minimization. The fact that in the nuclear case the ex- the system for a narrow energy interval around any given pansion can be safely stopped at the surface term while energy; that is the level density at that energy. Its loga- retaining accuracies of the order of one percent down to rithm is the entropy, now the “thermodynamic” potential sizes as small as A = 20 gives us the hope that just by in terms of V , N and E. That is all there is to do. However, adding a surface term we may obtain similarly adequate measurements typically deal with the canonical world of descriptions for many other mesoscopic systems. Further- temperature T and variables depending on T . more, for short range forces it happens that the surface Consequently one takes the derivative of entropy with coefficient is nearly equal and opposite to the volume co- respect to energy and calls that the inverse temperature efficient, i.e. a (T ) = −a (T ). This leads to a great deal ∂S 1 v s = β = . (2) ∂E T of economy and to an extreme generality. From this equation other quantities like the caloric curve It is important to appreciate that in any calculation ∂E E = E(T ) and specific heat CV = ∂T |V are determined. applied to mesoscopic systems, the presence of the sur- But it is here that problems arise. Equation (2) is a macro- face as a physical boundary, and boundary condition is of scopic thermodynamic relationship. But what is its mi- paramount importance for all calculations. Many of the croscopic origin? And does it apply all the time? In the claimed novel and general features resulting from finite following we consider this in detail. system calculations are either strongly dependent on such boundary conditions, or are artifacts of unwittingly intro- 2.1 The transformation from energy scale to duced boundary conditions which may not have necessar- temperature scale ily any specific physical meaning. We shall exemplify the above ideas by considering phase We are going to perform this transformation in two equiva- coexistence and phase transitions. lent ways: the first more physical, the second more formal. Let us consider a microcanonical system fully charac- terized by the relationship 2 Microcanonical and canonical worlds and ρ = ρ(E),S = S(E). (3) their languages Let us put a small system (thermometer) with a few de- The microcanonical world is conceptually the simplest grees of freedom (e.g. a classical one dimensional harmonic one. It deals with an ensemble of totally isolated systems. oscillator) into contact with the microcanonical system. 4 L. G. Moretto et al.: Mesoscopy and thermodynamics Any energy state ε of the thermometer is associated The independent variable β is the inverse of the canonical with a probability p (ε) equal to the degeneracy of the temperature T . In the canonical world at constant β (T ), microcanonical system at energy E − ε the microcanonical world at constant E is represented as a weighted sum, the weight being a Boltzmann factor. p(ε) ' ρ(E − ε)ρT (ε) (4) Equation (9) transforms the microcanonical world of where the level density of the one dimensional harmonic energy E and level density ρ(E) in to the canonical world oscillator ρT (ε) is constant. Given the “smallness” of the 1 of temperature T = β and partition function Z(β). This thermometer, ε is microscopically small compared to the transformation is a Laplace transform. “macroscopic” energy E. So the following Taylor expan- Notice that for any given T or β there is a distribution sion is in order of energies, and not a single value. The distribution in E 2 ∂ ln ρ 1 2 ∂ ln ρ ln p(ε) = ln ρ(E)− ε + ε 2 −· · · . (5) is the integrand of Eq. (9) ∂E ε=0 2 ∂E ε=0 But (E − E )2 p(E) = ρ(E) exp(−βE) = A exp − 0 .
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