Thermodynamically reversible processes in statistical physics John D. Norton
Citation: American Journal of Physics 85, 135 (2017); doi: 10.1119/1.4966907 View online: http://dx.doi.org/10.1119/1.4966907 View Table of Contents: http://aapt.scitation.org/toc/ajp/85/2 Published by the American Association of Physics Teachers
Articles you may be interested in Sliding down an arbitrary curve in the presence of friction American Journal of Physics 85, (2017); 10.1119/1.4966628
Kepler and the origins of pre-Newtonian mass American Journal of Physics 85, (2017); 10.1119/1.4972044
Weight of an hourglass—Theory and experiment in quantitative comparison American Journal of Physics 85, (2017); 10.1119/1.4973527
The Quantum Handshake: Entanglement, Nonlocality, and Transactions American Journal of Physics 85, (2017); 10.1119/1.4966111
QUANTUM MEASUREMENTS American Journal of Physics 85, (2016); 10.1119/1.4967925
Resource Letter HCMP-1: History of Condensed Matter Physics American Journal of Physics 85, (2017); 10.1119/1.4967844 Thermodynamically reversible processes in statistical physics John D. Norton Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 (Received 29 November 2015; accepted 19 October 2016) Equilibrium states are used as limit states to define thermodynamically reversible processes. When these processes are understood in terms of statistical physics, these limit states can change with time due to thermal fluctuations. For macroscopic systems, the changes are insignificant on ordinary time scales and what little change there is can be suppressed by macroscopically negligible, entropy-creating dissipation. For systems of molecular sizes, the changes are large on short time scales. They can only sometimes be suppressed with significant entropy-creating dissipation, and this entropy creation is unavoidable if any process is to proceed to completion. As a result, at molecular scales, thermodynamically reversible processes are impossible in principle. Unlike the macroscopic case, they cannot be realized even approximately when we account for all sources of dissipation, and argumentation invoking them on molecular scales can lead to spurious conclusions. VC 2017 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4966907]
I. INTRODUCTION results are then illustrated in Sec. IV with the case of an isothermal expansion of an ideal gas. In ordinary thermodynamics, a reversible process is, loosely speaking, one whose driving forces are so delicately II. SELF-CONTAINED THERMODYNAMICALLY balanced around equilibrium that only a very slight distur- bance to them can lead the process to reverse direction. REVERSIBLE PROCESSES Because such processes are arbitrarily close to a perfect bal- If our treatment of thermodynamically reversible pro- ance of driving forces, they proceed arbitrarily slowly while cesses is to be consistent, then we must consider the thermal their states remain arbitrarily close to equilibrium states. and statistical properties of all the components involved in They can never become equilibrium states, for otherwise the process. This may seem like a minor point, however, there would be no imbalance of driving forces, no change, fully implementing it is essential to all that follows. A full and no process. Equilibrium states remain as they are. implementation is rare since many common goals can be met This circumstance changes when we allow that thermal without it. We may merely wish, for example, to determine systems consist of very many interacting components, such the thermodynamic properties of some system, such as the as molecules, whose behavior is to be analyzed statistically. volume dependence of the entropy of a gas. Then we can Then what were the limiting equilibrium states of ordinary take shortcuts. thermodynamics are no longer unchanging. Molecular scale In a common case of the shortcut, the gas is confined to a thermal fluctuations—thermal noise—move them to neigh- cylinder under a weighted piston and the entirety of the sys- boring states and, since there are no directed imbalances of tem is within a heat bath that maintains all components at a driving forces, these migrations meander indifferently in a fixed temperature T. Following a familiar textbook treat- random walk. The very slight imbalance of forces of a ment,1 the piston is weighted by a pile of sand whose mass is reversible process must overcome this meandering if the pro- just enough to balance the gas pressure. No process will cess is to complete. On macroscopic scales, the fluctuation-derived meander- ensue, unless something changes. Tiny grains of sand are ing is negligible and what little there is can easily be over- removed, one by one, successively lightening the load on the come by very slight imbalances in the driving forces. On piston. With each removal the gas expands slightly and the molecular scales, however, fluctuations are large and signifi- gas pressure drops slightly until the slightly less weighty pis- cant imbalances in the driving forces are needed to bring any ton once again balances the pressure. Repeated removals process to completion. Because such imbalances are dissipa- realize a thermodynamically reversible expansion of the tive, creating entropy, reversible processes are impossible on confined gas. The entropy change in the gas DS can now be molecular scales. Completion of a process is only assured determined by tracking the heat Qrev Ðgained by the gas, probabilistically, with higher probabilities requiring greater according to the Clausius formula DS ¼ dQrev/T. entropy creation. In common treatments of thermodynamically reversible The principal goal of this paper is to demonstrate these processes in statistical physics, all details of the machinery last claims at the general level and to provide an illustra- that slowly carries the process forward are omitted. In its tion of them in the isothermal expansion of an ideal gas. place is the abstract notion of the manipulation of a variable, Section II introduces the essential but neglected idea that such as the volume of the expanding gas. The variable may one cannot properly assess the dissipation associated with be identified as an “external parameter” whose manipulation a process unless one accounts for all sources of dissipation. comprises a “switching process,”2 or as a “control parame- For reversible processes, that includes the normally sup- ter” that is “controlled by an external agent.”3 pressed devices that guide the process in its slow advance. In assuming that the external agent can slowly advance Section III contains the main results for the cases of pro- the control parameter, these reduced treatments neglect dissi- cesses in both isolated and in isothermal systems. These pation in the physical processes implementing the external
135 Am. J. Phys. 85 (2), February 2017 http://aapt.org/ajp VC 2017 American Association of Physics Teachers 135 manipulation. It is assumed tacitly, for example, that the @Fsys dFsys mechanism that lightens the load on the piston can be imple- X ¼ ¼ : (3) @x T dk mented in some reversible, non-dissipative manner that is consistent with the fuller thermodynamic and statistical An equivalent formulation of Eq. (2) is theory. In principle, an explicit determination of compatibility of X ¼ 0: (4) the process with our fuller theory would require examina- tions of the details of the external agent’s physical processes. The most familiar example of such a generalized force is Just what are the details of the non-dissipative machinery pressure P, and its associated displacement variable is vol- that picks off the sand grains one at a time? Only then have ume V. For a reversible expansion of a gas, the total general- we shown that the process is theoretically self-contained; ized force will be the suitably formulated sum of the that is, it relies only on the components manifestly conform- pressure force of the gas and the restraining forces on the pis- ing to our thermodynamic and statistical theory. ton that hold the system in equilibrium. They will sum to For macroscopic systems, neglecting these details is zero, as required in Eq. (4). usually benign, especially if our concern is merely comput- ing thermodynamic properties. The need to attend to these B. Limit states in statistical physics details becomes acute when we investigate processes on If a system is in one of the limiting equilibrium states of molecular scales, for fluctuations within molecular scale Eqs. (1) and (2) of ordinary thermodynamics, it is unchang- machinery are large and can disrupt the intended operation. ing. If we allow for its molecular constitution, then the equi- As we shall see, entropy creating disequilibria are required librium is dynamic with its components interacting under the to overcome the fluctuations and bring any process in a Hamiltonian evolution of a phase space. Through this inter- molecular-scale device to completion. nal dynamics, these states—now just called “limit states”— The discussion that follows is limited to self-contained are no longer unchanging. They can migrate to neighboring thermodynamically reversible processes, since these are the states through what manifests macroscopically as thermal only processes fully licensed by thermodynamic and statisti- fluctuations. We will consider two cases. cal theory. First, consider an isolated system. It is microcanonical; that is, its probability density is uniform over its classical III. THERMODYNAMICALLY REVERSIBLE phase space. As it migrates over the phase space, the proba- PROCESSES: GENERAL RESULTS bility that the system is in some region of the phase space is proportional to its phase volume A. Limit states in ordinary thermodynamics probability / phase volume: (5) In ordinary thermodynamics, a thermodynamically revers- ible process is one whose states come arbitrarily close to lim- System states can be associated with regions of the phase iting equilibrium states. For isolated systems, the equilibrium space. The entropy S assigned to them is states approached have constant thermodynamic entropy. We can track the progress of the process by a general parameter k, S / k ln ðphase volumeÞ; (6) such as the volume V of an expanding gas or the temperature T of a cooling system. If the process progresses from an initial where k is Boltzmann’s constant. Combining these equa- value kinit to a final value kfin, we have for the total entropy tions, we have Stot of the system that S / k ln ðprobabilityÞ or probability / exp ðS=kÞ: (7)
dStot=dk ¼ 0 and Einstein6 called Eq. (7) “Boltzmann’s principle” when he StotðkinitÞ¼::: ¼ StotðkÞ¼::: ¼ StotðkfinÞ: (1) introduced it in his analysis of fluctuations. It tells us that isolated thermal systems can fluctuate from high to low An important special case is an isothermal reversible pro- entropy states, but only with very small probability. cess, where the subsystem “sys” is maintained at a constant Second, consider a system in a heat bath, with which it temperature T by heat exchange with a heat bath environ- exchanges heat but no work, and is maintained by the bath at ment “env,” with which it exchanges no work. For this pro- constant temperature T. The system will be canonically dis- cess, the constancy of total entropy in Eq. (1) is equivalent to tributed over its phase space. That means that the probability the constancy of the free energy F ¼ U – TS of the system, 4 density of finding the system at a phase point with energy E, where U is the internal energy in the course of its migration over the phase space, is propor- tional to exp(–E/kT). Hence, the probability that it is found dFsys=dk ¼ 0 and in some subvolume Vph of its entire phase space is propor- FsysðkinitÞ¼::: ¼ FsysðkÞ¼::: ¼ FsysðkfinÞ: (2) tional to the partition integral Z(Vph), so that ð A generalized force X and associated displacement variable probability / ZðVphÞ¼ exp ð–E=kTÞdX; (8) x are defined so that the amount of work done dW by the sys- Vph tem in a small constant-temperature change is dW ¼ Xdx.If X is the total generalized force and we use the displacement where dX is the phase space volume element.7 A thermal variable x to track the degree of completion of the process, state is just the set of phase points that realize it. Thus the so that x ¼ k, then X is given by5 partition integral of Eq. (8) over the volume of phase space
136 Am. J. Phys., Vol. 85, No. 2, February 2017 John D. Norton 136 Vph containing these points gives the state’s free energy F, exchanges with its thermal surroundings. If the system is in through the canonical formula one of the limit states of Eq. (1) or Eq. (2) of some reversible process, fluctuations will bump the system to neighboring F ¼ –kT ln Z: (9) equilibrium states and then on still further to other neighbor- ing equilibrium states, and so on, in a migration that can Combining these equations, we have proceed over all the equilibrium states of the process. The equilibrium states have become pseudo-equilibrium states, F / –kT ln ðprobabilityÞ or probability / exp ð–F=kTÞ: confounded by fluctuations. The fluctuation-induced migra- (10) tion will eventually lead the system to occupy all the limit states with equal probability, as shown in Eq. (11). The limit states of a reversible process in an isolated sys- With macroscopic systems, the migration can be neglected tem represented in Eq. (1) have equal entropy S. It follows because the time scales needed to realize it are enormous. The from Eq. (7) that thermal fluctuations can bring the system pea in quiescent broth mentioned above illustrates the time spontaneously to any of the limit states with equal scale. It will eventually migrate over the entire bowl, but not probability in our lifetimes. With molecular-scale systems, the migration Pðk Þ¼… ¼ PðkÞ¼… ¼ Pðk Þ: (11) will be rapid and completely disrupt the intended reversible init fin process. We may initiate a molecular-scale process in or very This result of equal probability also holds for the limit states near to some state corresponding to kinit and then expect that of a reversible process, represented in Eq. (2) in an isother- the system will very slowly migrate through the states of mal system, for each state has equal free energy F and thus intermediate k values, terminating in that of kfin.However, in Eq. (10) equal probability. thermal fluctuations will defeat these expectations and move A familiar illustration of Eq. (11) is provided by a micro- the system rapidly among all the states. Termination will be scopically visible Brownian particle suspended in water in a impossible. If the system occupies a state at or near that of dish. If k is the position of the particle as it moves about, kfin, fluctuations will immediately divert it to other, earlier then each k state has equal entropy S (if the dish is isolated) states in the process. Thermodynamically reversible processes or equal free energy F (if the dish is in a heat bath). Over on molecular scales are impossible. time, as it executes a random walk, the Brownian particle will visit each position, and according to Eq. (11), with equal D. Dissipation suppresses fluctuations probabilistically probability. The time needed to realize these motions depends on the scale. For smaller Brownian particles, as their Once we allow that the limiting states are in pseudo- size approaches molecular scales, the motions become rapid, equilibrium, we see that an attempt at a reversible process comparable to those of individual water molecules. For can only be brought to completion if we introduce some dis- larger particles, approaching macroscopic sizes, the motions sipative, entropy-creating disequilibrium or imbalance of become so slow as to be negligible. A pea suspended in qui- forces that overpowers the fluctuations. The process of com- escent broth will eventually explore the complete bowl pressing a gas can be driven to completion by a piston whose through its Brownian motion, but its migration will require weight is sufficiently great to overpower the pressure of the eons and be undetectable on all normal time scales. gas, even allowing for fluctuations in the pressure. The dissi- Allowing for the statistical character of the limiting equi- pation replaces the uniform probability distribution, Eq. (11), librium states of a thermodynamically reversible process by one that favors completion, which can only be assured to thus reveals that they are no longer equilibrium states. some nominated probability. That is, we set the ratio P(kfin)/ Rather, they are pseudo-equilibrium states in the sense that P(kinit), which determines how much more likely the system they are no longer unchanging and can migrate spontane- is to settle into the final state kfin as opposed to reverting by ously through thermal fluctuations to other states. In macro- fluctuations to the initial state kinit. The corresponding dissi- scopic applications, this pseudo-equilibrium character can be pation is computed through Eqs. (7) and (10). For an isolated ignored because the time scales needed for it to manifest are system, the entropy change DS between initial and final enormous. On molecular scales, this pseudo-equilibrium states is character can no longer be ignored. DS ¼ k ln ½PðkfinÞ=PðkinitÞ or
C. Fluctuations make reversible processes impossible on PðkfinÞ=PðkinitÞ¼exp ðDS=kÞ: (12) molecular scales To be a reversible process in ordinary thermodynamics, For a system in a heat bath at temperature T with which it the states of the process must come arbitrarily close to limit exchanges no work, the free energy change DF between ini- states. As they do so, the states become ever more delicately tial and final states is balanced. In ordinary thermodynamics, these limit states are D – k = k equilibrium states and there are no disturbing forces present F ¼ kT ln ½Pð finÞ Pð initÞ or to upset the delicate balance. PðkfinÞ=PðkinitÞ¼exp ð–DF=kTÞ: (13) This is no longer so once we allow for the statistical char- acter of the limiting states. Thermal fluctuations in statistical These equations apply to a system that is initially set up in physics provide disturbing forces. They bump a thermal sys- state kinit, then released and allowed to equilibrate. We write tem to neighboring states. Volume fluctuations lead a gas P(kfin) for the probability that it will subsequently be found volume to expand and contract slightly, and energy fluctua- in state kfin. The probability P(kinit)isnot the probability tions lead the gas to heat and cool slightly through energy that the system was initially set up in state kinit. It is the
137 Am. J. Phys., Vol. 85, No. 2, February 2017 John D. Norton 137 probability that the system, after achieving its new equilibra- If our intended process is the migration of a Brownian par- tion, reverts by a fluctuation to the initial state. ticle from one side of a dish to the other, the entropy- These two formulas, Eqs. (12) and (13), do not give the creating disequilibrium needed to suppress fluctuations is total entropy and free energy changes directly for most pro- introduced by inclining the dish so that the Brownian particle cesses. Commonly, processes can only arrive at the final is driven in the intended direction by gravity. state if many other intermediate states are also accessible, The quantities of entropy produced and the associated such as the intermediate states of the expansion of a gas. probabilities of completion are computed in the supplemen- Their accessibility leads to further dissipative creation of tary material.10 The supplementary material also illustrates a entropy or further free energy decreases. Because these inter- simple way in which the intermediate states can be made mediate states remain accessible, this further dissipation probabilistically inaccessible, in order to arrive at the case of must be included in the computation of the total dissipation. minimum dissipation. To arrive at the minimum dissipation, all of these other inter- mediate states, incompatible with the initial and final states, IV. SELF-CONTAINED, ISOTHERMAL EXPANSION must be rendered highly improbable by careful design of the OF A IDEAL GAS process. That is achievable but not done in most standard processes. If we do contrive the process so that the initial The general results of Sec. III can be illustrated in the case and final states only are accessible,8 then of a self-contained, reversible, isothermal expansion of an ideal gas. For the results of Sec. III to apply, the analysis PðkinitÞþPðkfinÞ¼1: (14) must include the mechanism through which the expanding gas is kept in near perfect equilibrium with the restraining With this contrivance, the minimum entropy creation in an 9 piston. If that mechanism is the device of Sec. II that removes isolated system is sand grains one at a time, its operation would have to be ana- lyzed for dissipative processes. This analysis would be com- PðÞkinit þ PðÞkfin PðÞkfin DSmin ¼ k ln ¼ k ln 1 þ : plicated; it would also be unnecessary, since there are simpler PðÞkinit PðÞkinit ways of achieving the same effect of a self-contained process. (15) One way is to replace the homogeneous gravitational field acting on the piston by another, inhomogeneous field. It For a system in a heat bath at temperature T with which it weakens as the piston rises by just the amount needed to exchanges no work, the minimum free energy change in exe- maintain a mechanical balance of forces, without any manip- cuting the process is ulation of the weighting of the piston itself.11 Another approach is presented in detail below. Through a simple PðÞk þ PðÞk DF ¼ kT ln init fin mechanical contrivance described in Sec. IV K, the piston min PðÞk init area increases as the gas expands in such a way that the total PðÞkfin upward force exerted by the gas on the piston remains con- ¼ kT ln 1 þ : (16) stant, balancing the constant weight of the piston. PðÞkinit
A modest probability ratio for success is A. The confined gas and the stages of its expansion An ideal gas of n monatomic molecules is contained in a Pðk Þ=Pðk Þ¼20; for which DS ¼ 3k and fin init chamber under a horizontal, weighted piston in a heat bath DF ¼ –3kT: (17) that maintains the system of gas and piston at a constant tem- perature T. The gas expands reversibly by raising the piston, In molecular scale systems, a dissipation of entropy 3 k and passing work energy to the rising weight. The expansion is free energy 3kT is comparable to the entire amounts of made self-contained by ensuring that the piston area A(h)of entropy and free energy changes. It is a significant departure the piston at height h increases by just the right amount that from equilibrium. Thus the conditions for completion of a the weight of the piston always balances the mean pressure thermodynamically reversible process cannot be met at force of the gas for the limiting states. The expansion begins molecular scales: completion requires that the system not with the piston at h h when the gas has spatial volume approach the limit states too closely, which entails that the ¼ 0 V(h ) and ends at h h with gas spatial volume V(h ). process cannot be thermodynamically reversible. 0 ¼ 1 1 The stages of the process of expansion are, loosely speak- For macroscopic systems with component numbers of the 23 ing, parameterized by the height to which the gas has lifted order of Avogadro’s number N ¼ 6.022 10 , quantities of entropy are of the order of Nk and quantities of free energy the piston. This is not precisely correct, however, because of NkT. The dissipation required is negligible. If completion the fluctuating thermal energy of the piston will allow it to is required with very high probability, we might choose the rise above the maximum extension of the gas. We shall see ratio that this effect is negligible for a macroscopic gas, but is cer- tainly noticeable for a gas of one or few molecules. To 10 PðkfinÞ=PðkinitÞ¼7:2 10 ; for which accommodate this effect, the limiting equilibrium states DS ¼ 25 k and DF ¼ –25 kT: (18) associated with the expansion are parameterized by the height h above the chamber floor that demarcates the region This level of dissipation is still insignificant for macroscopic accessible to the gas and the region accessible to the piston. systems. Thus, molecular-scale dissipation provides no That is, if the height of the ith molecule is given by xi and obstacle to thermodynamically reversible processes at mac- the height of the piston by xpist, then the limiting equilibrium roscopic scales. states are characterized by
138 Am. J. Phys., Vol. 85, No. 2, February 2017 John D. Norton 138 0 < xi < h; for all i and xpist h: (19) xpist > xi; for all i; (21) and The resulting “h-states” are not completely disjoint in the X sense that two may share some of the same microstates. For E ðx; pÞ¼ p2=2m; example, states h and 2 h may share the same microstate as gas i¼1;n follows. In state h, a thermal fluctuation may abruptly push 2 the piston to height 2 h while leaving all the gas molecules Epistonðxpist; ppistÞ¼ppist =2M þ Mgxpist: (22) below height h. The same microstate may be associated with The constant g is the gravitational field strength; it is state 2 h if all the gas molecules unexpectedly collect below assumed that the individual molecules do not feel the gravi- height h through a thermal fluctuation. tational force acting on the piston. This example makes clear that an extensive overlap of the The condition xpist > xi asserts that the piston never falls to microstates attached to h-states is improbable for a macro- or below the height of the highest molecule. It expresses the scopic gas of large n. For, as we shall see in calculations coupling between gas and piston. The fact of this coupling below, large volume fluctuations are extremely improbable would normally mean that the gas-piston partition function in the short term. Correspondingly, for large n, the mass of does not factor. However, the h-state of Eq. (19) has the the piston will be great, so that the spatial extent of its short- fortunate property of breaking the coupling for each fixed term fluctuations will be small. However for a gas of one or value of h, so that the gas-piston partition integral for state h, few molecules, the fluctuations will be large in relation to Zgas-piston(h), is the product of the partition integrals for the the system size. As a result, a single microstate, specified by individual gas and piston systems the position of the gas molecules and piston, can correspond to a wide range of h-states. This ambiguity in the h-states is Zgas-pistonðhÞ¼Zgas ðhÞ ZpistonðhÞ; (23) part of the breakdown of reversible processes at molecular scales; there is a failure of distinctness of the individual and their free energies F, as given by the canonical formula stages through which we would like to the process to pass. F ¼ –kT ln Z, will sum Figure 1 illustrates how h-states for heights h and 2 h are F h F h F h : (24) almost certainly realized by distinct microstates, if the gas is gas-pistonð Þ¼ gas ð Þþ pistonð Þ macroscopic. However, just one microstate can realize both Thus, we can compute the thermodynamic properties of the h-states for a gas of very few molecules. gas and piston independently for these states.
B. Gas-piston Hamiltonian C. Gas properties The ideal gas is composed of n monatomic molecules, The gas partition integral is each of mass m. The canonical position coordinates of the molecules are collected in the vector x, whose 3 n compo- ZgasðÞh nents ({xi}, {yi}, {zi}), where i ¼ 1,…,n, are the Cartesian ð EðÞx; p position coordinates of each of the n molecules, and the ¼ exp dx dp canonical momenta are similarly collected in the vector p, kT all x;p whose 3 n components are the corresponding momentum Y ð p 2 þ p 2 þ p 2 components of each of the molecules. The piston of mass M ¼ exp x;i y;i z;i 2mkT has two relevant degrees of freedom, its vertical canonical i¼1;n all pi ð ðð position xpist and its vertical canonical momentum ppist. The Y h combined Hamiltonian of the gas-piston system is dpx;i dpy;i dpz;i dxidyidzi i¼1;n xi¼0 accessible yi;zi ð Egas pistonðx; p; xpist; ppistÞ¼Egasðx; pÞ Y h - 3n=2 ¼ ðÞ2pmkT AxðÞi dxi þ E ðx ; p Þ; (20) piston pist pist i¼1;n xi¼0 3n=2 n where we require in Eq. (20) that ¼ ðÞ2pmkT VhðÞ; (25)
Fig. 1. For gases of very many molecules, two different h-states have distinct molecular microstates (first and second cylinders). For gases of very few mole- cules, two different h-states can have the same molecular microstates (third and fourth cylinders).
139 Am. J. Phys., Vol. 85, No. 2, February 2017 John D. Norton 139 where A(xi) is the gas chamber cross-sectional area at height dVðÞ h Mg AhðÞ¼ ¼ VhðÞ; (32) xi and V(h) is the spatial volume accessible to the gas mole- dh nkT cules between the chamber floor and height h. The canonical free energy is for h0 < h < h1. The solution is FgasðhÞ¼–kT ln ZgasðhÞ¼–nkT ln VðhÞþconstgasðTÞ; MgðÞ h h VhðÞ¼ VhðÞexp 0 (33) (26) 0 nkT where constgas(T) is a constant independent of h. Because V and is a monotonic function of h, we can use it as the path param- eter k to define the generalized force dVðÞ h Mg MgðÞ h h0 AhðÞ¼ ¼ VhðÞ0 exp dh nkT nkT @ nkT X ðÞV ¼ F ðÞV ¼ : (27) gas gas MgðÞ h h0 @V T V ¼ AhðÞexp : (34) 0 nkT That is, the generalized force is just the ordinary pressure of the gas according to the ideal gas law. Equations (32) and (34) tell us that the gas volume and pis- ton area must each grow exponentially with height h during D. Piston properties the expansion h0 < h < h1 for equilibrium to be maintained. The probability P(h) of each h-state is proportional to The piston partition integral is the partition integral Zgas-piston(h) ¼ Zgas (h).Zpiston(h), and is ð given as EpistonðÞx;p ZpistonðÞh ¼ exp dxdp kT PhðÞ/ Zgas-pistonðÞh allp;x ð ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 3n=2 np kT Mgh p Mgx ¼ ðÞ2pmkT VhðÞ 2pMkT exp ¼ exp dp exp dx Mg kT allp 2MkT x¼h kT pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kT Mgh 3n=2 n kT ¼ 2pMkT exp : (28) ¼ ðÞ2pmkT VhðÞ0 2pMkT Mg kT Mg Mgh exp 0 ¼ constðÞ T ; (35) The canonical free energy is kT
FpistonðhÞ¼ kT ln ZpistonðhÞ¼Mgh þ constpistonðTÞ; where Eq. (33) was used to show (29) n Mgh where const (T) is a constant independent of h. Using V VhðÞexp piston kT as the path parameter, the generalized force is n n MgðÞ h h0 ¼ VhðÞ0 exp @ nkT XpistonðÞV ¼ FpistonðÞV @V T MgðÞ h h0 Mgh0 exp exp @ dh Mg kT kT ¼ FpistonðÞh ¼ : (30) @h dVðÞ h AhðÞ Mgh T ¼ VhðÞn exp 0 : (36) 0 kT This is the ordinary gravitational force exerted per unit area by the weight of the piston. That is, Eq. (35) shows that each of the h-states is equally probable. It also follows from Eq. (35) that the free energy E. Balance of forces of each of these states is the same.
During the expansion, the piston rises from height h ¼ h0 to h ¼ h1. Associated with each height is a limit state in F. Fluctuations negligible for a macroscopic gas which the mean gas pressure force and piston weight are with large n equal, in the correlate of the equilibrium of ordinary thermo- dynamics. We recover this equality from the condition for In the h-state of Eq. (19), the mean gas pressure is bal- equilibrium: the free energy of the gas-and-piston system anced precisely by the weight of the piston. Fluctuations will remains constant as in Eq. (2), or, equivalently, that the total lead the gas pressure force sometimes to exceed and some- generalized force vanishes as in Eq. (4). Setting the sum of times to be less than the piston weight. As a result, the sys- the generalized forces of Eqs. (27) and (30) to zero, we have tem will migrate up or down to neighboring, equally probable h-states. For a macroscopic gas, however, the nkT Mg migration will be so slow that it will not manifest on ordi- ¼ 0: (31) VhðÞ AhðÞ nary time scales. To see this, recall that the motions that lead to the migra- Since A(h) ¼ dV(h)/dh, this last condition leads to the differ- tion of the piston are due to the thermal fluctuations in the ential equation piston. The piston will have equipartition energy of kT/2 in
140 Am. J. Phys., Vol. 85, No. 2, February 2017 John D. Norton 140 1 2 its kinetic energy 2 Mvrms, where vrms is the piston root- Therefore, a reversible gas expansion is quite achievable in mean-square velocity. A liter of an ideal gas forms a cube of the sense that its states can be brought arbitrarily close by side 10 cm with a piston area of 100 cm2. At one atmosphere macroscopic standards to the equilibrium states. Nonetheless, pressure, or 1.033 kg/cm2 in engineering units, the piston just as in the case of the Brownian motion of a macroscopic 1 2 mass M is therefore 103.3 kg. Solving 2 Mvrms ¼ (1/2)kT at body, tiny fluctuations will accumulate over long times and ˚ ˚ 10 25 C, we find vrms ¼ 0.0631 A/s. Since 1 A ¼ 10 m is ten eventually enable the gas-piston system to migrate over the orders of magnitude smaller than macroscopic scales and full extent of configurations available to it. This migration is this tiny speed will not be sustained unidirectionally more represented by the equal probabilities of all states of Eq. (11). than momentarily, the h-state is, on ordinary time scales and at macroscopic length scales, a quiescent state. G. Fluctuations for n 5 1 Another way to see that fluctuations are negligible for macroscopic systems is to look at the fluctuations in each of Matters change when we take small values of n. The the gas and piston systems taken individually. If we assume extreme case of a one-molecule gas is dominated by fluctua- that the piston is confined to heights h H but otherwise tions. The formulae developed above still apply. However, free, its positions will be Boltzmann-distributed probabilisti- we must now set n ¼ 1 in them. In place of Eq. (39), we have cally according to a probability density over heights h a piston whose thermal fluctuations fling the piston through distances of the order of the size of the entire gas Mg MgðÞ h H p ðÞh ¼ exp ; (37) piston kT kT ðsize of piston position fluctuationsÞ¼ðlinear size gasÞ: (41) for h H. This is an exponential distribution for which It is also evident without calculation that a gas of a single mean ¼ standard deviation ¼ kT=Mg: (38) molecule is undergoing massive density fluctuations as the molecule moves from region to region. If we associate the Thus, kT/Mg is a measure of the linear size of the volume of a gas with the places where its density is high, fluctuation-induced displacements of the piston from its floor these in turn can be understood as volume fluctuations of the height H. size of the gas-confining chamber This measure is very small in comparison with the overall linear size of the gas-piston system. A convenient measure ðsize of gas volume fluctuationsÞ’ðgas volumeÞ: (42) of the linear size of the gas is the ratio V(H)/A(H). If the gas is confined to a cubical box, this ratio is the length of the That fluctuations will dominate is apparent from brief side. We find directly from Eqs. (32) and (38) that reflections without calculations. It is assumed that the pres- sure of the one-molecule gas is sufficient to support the ðsize of piston position fluctuationsÞ¼ðlinear size gasÞ=n: weight of the piston. That is, in molecular terms, repeated (39) collisions with a single rapidly moving molecule are enough to support the mass of the piston. This can only be the case if For macroscopic samples of gases, n will be of the order of 23 the piston mass itself is extremely small. If that is so, then its Avogadro’s number N ¼ 6.022 10 . Hence, the fluctuation- own thermal motion will be considerable. induced disturbance to the equilibrium limit state will be neg- These fluctuations defeat attempts to realize a thermody- ligible. For example, a liter of an ideal gas at 25 Candone namically reversible expansion of a gas of one or few mole- atmosphere pressure forms a cube of side 10 cm and contains 22 cules. In such an expansion, the gas state is always 2.46 10 molecules. According to Eq. (39), the linear size arbitrarily close to the limit states and it is supposed to of the fluctuations is 10/(2.46 1022)cm 4.065 10 14 A˚ . ¼ migrate indefinitely slowly through them, under the delicate That is, the size of the fluctuations is roughly three orders of and very slight imbalance of pressure and weight forces. magnitude smaller than atomic sizes. This circumstance is unrealizable. The fluctuations just Consideration of volume fluctuations in the gas yields described will completely destabilize the delicate imbalance. similar negligible deviations. The probability that an ideal gas of n molecules of volume V fluctuates to a smaller vol- If the gas-piston system has arrived at any height, fluctua- ume V – DV is [(V – DV)/V]n. Because n is so large, this tions will immediately move it to a different height. A nearly probability can only appreciably differ from zero if DV/V is completed expansion may be flung back to the start of the very small, so that [(V – DV)/V]n [1 – n(DV/V)] ¼ [1 – DV/ expansion, just as an unexpanded gas can be rapidly (V/n)]. This probability will still only appreciably differ from expanded by a fluctuation. Instead of rising serenely, the pis- zero if the magnitude of the fluctuations DV is of the order of ton will jump about wildly with no discernible start or finish V/n or smaller; that is, to the process.
ðsize of gas volume fluctuationsÞ < ðgas volumeÞ=n: H. Suppressing fluctuations: A rough estimate (40) An assured expansion, not confounded by fluctuations, The h-state of Eq. (19) does not represent perfectly the inter- will only be possible if we introduce enough entropy- mediate states of the gas expansion, since fluctuations in gas creating disequilibrium to suppress the fluctuations. A very volume and piston position will breach the boundary at rough first estimate confirms that the amount of entropy that height h between the gas and the piston. However, these cal- must be created will be considerably greater than the entropy culations show that for macroscopic gases the breaches are change between the states of the unexpanded and the entirely negligible. expanded one-molecule gas. It will be negligibly small,
141 Am. J. Phys., Vol. 85, No. 2, February 2017 John D. Norton 141 however, in relation to the entropy change in the expansion 1 DS ¼ DS þ DS ¼ Mv2 =T: (48) of macroscopic gas. gas env 2 proc If the motion of expansion is to dominate the random ther- mal motions, then the vertical velocity of the piston in the Thus, the net increase in entropy results entirely from overall process must greatly exceed its random thermal the irreversible transfer of the potentially usable work as 1 2 velocity. Assume that the mass M of the piston is slightly heat Q ¼ 2 Mvproc to the environment, which creates smaller than the equilibrium value required in Eq. (32),so entropy Q/T. that there is a small, net upward force on the piston. This The random thermal motion of the piston is measured by upward force gradually accelerates the piston until, at the its root-mean-square vertical speed vtherm, which satisfies end of its expansion, it has acquired the vertical speed vproc and then slams to a halt. This process speed “proc” is a rough 1 1 Mv2 ¼ kT: (49) measure of the overall vertical motion of the piston. 2 therm 2 1 2 The associated kinetic energy 2 Mvproc is derived from work done on the piston. It is potentially usable work energy The condition that random thermal motions not confound the that is lost as heat to the environment at the conclusion of process is the process. Had the process been carried out non- dissipatively, that is, reversibly, the only difference in the vproc vtherm: (50) end state is that this lost work would have been stored as extra potential energy in the ascent of a weightier piston and It follows immediately from the two preceding equations the corresponding quantity of heat would not have been that passed irreversibly to the environment. The dissipation is represented most compactly in terms 1 1 DF kT and DS k: (51) of free energy. The free energy change of the gas-piston 2 2 system is On molecular scales, this decrease in free energy or increase DF ¼ DFgas þ DFpist of entropy represents a considerable dissipation and depar- ¼ DU T DS þ DU TDS : (43) ture from equilibrium. For comparison, the free energy and gas gas pist pist entropy changes usually attributed to a two-fold, reversible For a reversible, non-dissipative expansion, we have isothermal expansion of a one-molecule gas are just DF DF ¼ 0. Most of the terms in this expression remain the ¼ kT ln 2 ¼ 0.69 kT and DS ¼ k ln 2 ¼ 0.69 k. same if we now consider the dissipative expansion. The I. Suppressing fluctuations: Free energy changes internal energy Ugas and entropy Sgas of the gas are functions of state, so they remain the same. The entropy of the piston Lightening the piston so that vproc vtherm enables the is unaltered because it is just a raised mass, so DSpist ¼ 0. expansion to complete with dissipation corresponding to Overall, in the transition to a dissipative expansion, the free the free energy decrease and entropy increase of Eq. (51) energy change DF is depressed from its zero value merely with a reasonably high, but unquantified, probability. A by the decrease in DUpist below its reversible value in the 1 2 closer analysis using Eq. (13) provides quantitative rela- amount of the lost work 2 Mvproc. That is, we have tions among the amount piston mass lightening, the dissipa- tion, and the probability of completion. We will find that 1 2 negligible lightening and dissipation can assure completion DF ¼ Mvproc: (44) 2 with high probability for a macroscopic gas, but that no The dissipation can also be measured by an entropy amount of lightening can achieve this for a one-molecule change, but now we must consider the entropy of the gas, gas. If M is the equilibrium mass defined through Eq. (32), piston, and environment together. If DUenv,rev is the change eq of internal energy of the environment in the case of the then we introduce a slight disequilibrium by setting the pis- reversible process, then we have ton mass M to be slightly smaller 1 M ¼ M DM; (52) DU ¼ DU þ Mv2 : (45) eq env env;rev 2 proc where DM > 0. Instead of Eq. (35), we have for the probabil- Hence, the total entropy change in the environment is ities P(h) of the h-states
1 2 DSenv ¼ DUenv;rev=T þ Mvproc=T: (46) PhðÞ/ Z ðÞh 2 gas-piston ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3n=2 np kT Mgh Since the start and end states of the gas are the same for the ¼ ðÞ2pmkT VhðÞ 2pMkT exp reversible and the irreversible processes and entropy is a Mg kT ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi function of state, the entropy change in the gas is the same 3n=2 p kT ¼ ðÞ2pmkT VhðÞn 2pMkT for both processes. It follows that: 0 Mg DS ¼ –DU =T; (47) Mgh0 DMgðÞ h h0 gas env;rev exp exp ; (53) kT kT so that the total entropy change for gas, piston, and environ- ment together is since now
142 Am. J. Phys., Vol. 85, No. 2, February 2017 John D. Norton 142 ÀÁ n n Mgh n MeqghðÞ h0 Meq DM ghðÞ h0 Mgh0 VhðÞexp ¼ VhðÞ0 exp exp exp kT nkT kT kT Mgh DMgðÞ h h ¼ VhðÞn exp 0 exp 0 : (54) 0 kT kT
Most of the terms in Eq. (53) are independent of h, so it can PhðÞ1 MeqghðÞ1 h0 be re-expressed more usefully as12 ¼ exp PhðÞ0 DM!M kT eq n MeqghðÞ1 h0 PhðÞ1 DMgðÞ h1 h0 ZhðÞ1 DF ¼ exp ¼ exp ¼ ¼ exp : nkT PhðÞ0 kT ZhðÞ0 kT VhðÞn (55) ¼ 1 : (57) VhðÞ0 The free energy change DF between the two states is intro- duced using the canonical formula F ¼ kT ln Z. It follows This probability ratio is just the probability ratio associated that the free energy change is with a spontaneous recompression of the gas of n indepen- dently moving molecules from volume V(h1)toV(h0). DF ¼ –DMgðh1 h0Þ: (56) For gases of one or few molecules, the maximum of Eq. (57) presents serious problems. For the one-molecule gas This relation admits the obvious reading: in reducing undergoing a two-fold volume expansion, the largest proba- the piston mass by DM below the equilibrium mass Meq, bility ratio possible is just 2:1. Even in the most dissipative we lose the possibility of recovering work DMg(h1 h0) case, with the piston reduced to its lightest mass, the expand- when the piston is raised from height h0 to h1.That ing one-molecule gas is just twice as likely to be in the work would otherwise appear as a corresponding intended final state than in the initial state. increase in the potential energy of the unreduced piston In sum, a thermodynamically reversible expansion of a of mass M. gas of one or few molecules is impossible. Fluctuations We have already seen from Sec. III D that a macro- prevent the states of the expansion migrating very close to scopically negligible free energy change DF ¼ 25kT is and very slowly past the requisite sequence of pseudo- sufficient to ensure a very favorable probability of equilibrium states. In the system described, even dissipation completion. From Eq. (56), we see that this free energy in significant measure at molecular scales is unable to sup- change will correspond to a macroscopically negligible press the fluctuations. This in turn results from the limiting mass reduction. For a height difference of (h1 h0) pseudo-equilibrium states themselves being so confounded ¼ 10 cm and a gas at 300 K, the mass reduction is DM by fluctuations that they cannot persist even briefly as stable 19 ¼ 25kT/g(h1 h0) ¼ 1.05 10 kg, which is consider- states. ably less than the 103.3-kg piston mass of Sec. IV F. In sum, a thermodynamically reversible expansion of a K. How piston area increases macroscopic gas is possible in the following sense. The gas- piston system can expand slowly through a sequence of It is not so straightforward to devise ordinary mechanical states that are, by macroscopic standards, very close to limit devices that can achieve the increase of piston area required states that are stable in the shorter term. Fluctuations intro- by Eq. (34). The simplest arrangement, illustrated in Fig. 2, duce negligible complications. is to have a gas chamber of rectangular section that flares out horizontally in one direction with heights h > h0. The cham- ber is fitted with a horizontal, rectangular piston that J. Failure to suppress fluctuations for the one-molecule increases in area as it ascends, so it can keep the gas con- gas fined. The piston consists of two rectangular parts that slide The suppression of fluctuations breaks down completely, however, for a gas of one or few molecules. For the maxi- mum suppression is achieved by reducing the mass of the piston arbitrarily close to zero mass. That is, we achieve the maximum probability ratio favoring completion in Eq. (55) when DM approaches its maximum value Meq. This maxi- mum is the case of a massless piston, which is no piston at all; we are simply releasing the gas freely into an infinite space. Therefore, a canonical probability distribution is not established, and the probabilistic analysis used here does not apply. To preserve its applicability, consider instead the Fig. 2. A weighted piston remains in equilibrium with a gas during the gas limiting behavior as DM approaches Meq arbitrarily closely expansion, since the area of the piston increases as the expansion proceeds, but never actually equals Meq. Using Eq. (33) with Eq. (55), allowing the diminished gas pressure to exert the same upward force on the we have piston.
143 Am. J. Phys., Vol. 85, No. 2, February 2017 John D. Norton 143 without friction over each other and are guided apart by rails over the process stages. Such a process does not complete. as the piston ascends. Equations (12) and (13) determine the order of magnitude The sliding of the parts of the piston introduces new ther- of the dissipation needed to overcome the fluctuations and mal degrees of freedom. They can be neglected because they assure probabilistic completion of the intended process. are independent of the expansion. At all piston heights, each Equations (15) and (16) give the minimum dissipation in a sliding part has the same slight horizontal motion corre- special circumstance contrived to be least dissipative. The sponding to whatever slack is in the fitting of the rails to the dissipation is negligible on macroscopic scales and signifi- parts. Since this slack will be the same at all stages of the cant on molecular scales. expansion, they will contribute an additive term to the piston The idea that one could undertake a thermodynamically Hamiltonian that is independent of h and thus will not figure reversible expansion of a gas of a single molecule was intro- in the h dependence of the piston free energy of Eq. (29) or duced by Szilard13 as part of his celebrated analysis of in the generalized force of Eq. (30). Maxwell’s demon. The idea has become standard in the now Finally, the expansion under this scheme cannot continue voluminous literature that develops Szilard’s work.14 indefinitely. Otherwise the gas-piston system can access an Szilard15 briefly recognized the problem that the gas pressure infinity of equally accessible stages of expansion, which is wildly fluctuating, as it acts to lift a weight coupled to the means that it will never achieve equilibrium. The probability piston. The problem is dismissed with the parenthetically distributions used above, however, depend on the assumption inserted remark: that equilibrium has been achieved. The expansion could be halted by placing a maximum stop on the piston at some The transmission of force to the weight is best maximum height; however, this would introduce complicat- arranged so that the force exerted by the weight on ing thermal effects. As the piston approaches the stop, it the piston at any position of the latter equals the would behave like a one-molecule gas and resist compres- average pressure of the gas. sion. The simplest remedy is to assume that, at some height We have now seen here in detail that this is an inadequate Hmax h1, the chamber-piston system reverts to one with response. There is no arrangement that can convey the work constant piston area. Then achieving greater stages of expan- done by the expanding one-molecule gas to a raised weight sion ceases to be equally easy and an equilibration is in a way that maintains thermodynamic reversibility of the possible. entire process. Any arrangement, no matter how simple or complicated in design, is subject to the above general rela- V. CONCLUSION tions. They affirm that fluctuations will disrupt the intended operation, unless the fluctuations can be suppressed by the The accommodation of the molecular constitution of mat- dissipative creation of entropy in quantities significant at ter by ordinary thermodynamics introduces negligible com- molecular scales. plications for the thermodynamic analysis of macroscopic systems. However, as a matter of principle, once we take 1Such as in Hendrick C. Van Ness, Understanding Thermodynamics into account all the processes involved, thermal fluctuations (McGraw-Hill, New York, 1969; reprinted by Dover, New York, 1983), preclude thermodynamically reversible processes in systems pp. 19–22. at molecular scales. This has been shown in Sec. III for the 2Christopher Jarzynski, “Nonequilibrium equality for free energy differ- general case of any isolated system and for any system main- ences,” Phys. Rev. Lett. 78(14), 2690–2693 (1997). For a precise charac- tained at constant temperature by a heat bath with which it terization of thermodynamically reversible processes and a critical exchanges no work. historical survey, see John D. Norton, “The Impossible Process: In standard treatments of molecular-scale systems, ther- Thermodynamic Reversibility,” Studies in History and Philosophy of Modern Physics 55(2016), pp. 43–61. modynamically reversible processes are described as advanc- 3R. Kawai, J. M. R. Parrondo, and C. Van den Broeck, “Dissipation: The ing very slowly under the guidance of a parameter that is phase-space perspective,” Phys, Rev. Lett. 98, 080602 (2007). 4 manipulated externally by unspecified processes. The requi- To see this, note that for small changes we have dFsys ¼ dðUsys TSsysÞ site precise, external control of the parameter is only possible ¼ dUsys TdSsys ¼ dUsys þ dUenv ¼ dUtot ¼ 0, where the heat passed to through considerable dissipation in those unspecified pro- the environment in the reversible process is dQsys ¼ TdSsys, which equals cesses. It renders the overall process irreversible. The neglect the energy change in the environment dUenv. 5For a small, reversible change we have dF dU TdU dU of this additional dissipation masks the impossibility sys ¼ sys sys ¼ sys dQ ¼ dW, so that dFsys/dx ¼ dW/dx ¼ X. described here. 6Albert Einstein, “On a heuristic viewpoint concerning the production and The most general result is the impossibility of a reversible transformation of light,” Ann. Phys. 17, 132–148 (1905). process for any isolated, molecular-scale system, since it 7To connect with the usual statement of the canonical distribution, if covers all other cases. Imagine that somehow we could real- Vph,tot is the volume of the full phase space accessible to the system, ize a reversible process in some part of an isolated system. then the canonical distribution is p ¼ exp( E/kT)/ Z(Vph,tot) and the Since reversibility is unachievable for the total isolated sys- probability that the system is in subvolume Vph is equal to Z(Vph)/ Z(Vph,tot). tem, there must be an unaccounted dissipation in some other 8The intermediate states can never be completely inaccessible or the pro- part of the system. cess could not proceed. Rather the process design must be such as to make The impossibility of molecular-scale, thermodynami- them accessible only with arbitrarily small probability. 9 cally reversible processes derives from Eqs. (11)–(13) of Equations (15) and (16) with a term ln[1 þ P(kfin)/P(kinit)] give slightly Secs. III B and III D, which apply quite generally. If we higher dissipation than the corresponding formulas (22) and (23) of an ear- have a process that is intended to be thermodynamically lier paper [John D. Norton, “All shook up: Fluctuations, Maxwell’s reversible, Eq. (11) tells us that thermal fluctuations lead Demon and the thermodynamics of computation,” Entropy 15, 4432–4483 (2013)], which instead have a term ln[P(kfin)/P(kinit)]. The latter formulae the system to meander back and forth indefinitely if its presumed that the process ends in a way that prevents return to the initial states are in or arbitrarily near the limiting states. They state. In the absence of a non-dissipative way of preventing this return, the will eventually realize a uniform probability distribution newer formulae provide a better limit.
144 Am. J. Phys., Vol. 85, No. 2, February 2017 John D. Norton 144 10See supplementary material at http://dx.doi.org/10.1119/1.4966907 for Collected Works of Leo Szilard: Scientific Papers (MIT Press, “Appendix: Moving a Brownian particle”; also available at
145 Am. J. Phys., Vol. 85, No. 2, February 2017 John D. Norton 145