3 2 0 2 b e F 2 h c e m t a t s t a m d n o c 3 v 5 5 4 2 0 1 1 2 2 v i X r a A short derivation of Boltzmann distribution and Gibbs entropy formula from the fundamental postulate Laboratoire des solides irradi es Ecole polytechnique CEA CNRS IPP 91128 Palaiseau France D Lairez Dated February 3 2023 Introducing the Boltzmann distribution very early in a statistical thermodynamics course in the spirit of Feynmann has many didactic advantages in particular that of easily deriving the Gibbs entropy formula In this note a short derivation is proposed from the fundamental postulate of statistical mechanics and basics calculations accessible to undergraduate students Clausius entropy i e the state variable used in ther modynamics to account for heat exchanged and Gibbs entropy i e the one computed from the probability dis tribution pi of microstates are the same physical quan tity However this link and the famous formula1 S Xi pi ln pi 1 seem to be pulled out of a hat in most textbooks and undergraduate university courses These can be classi ed into categories lying between two main extreme ap proaches 1 Those who start by de ning statistical entropy with Eq 1 either without further justi cation e g in Huang2 or Reichl3 or by introducing the formula in Balian4 or Ben via information theory e g Naim and Casadei5 and then show that from a maximum entropy principle this makes it possi ble to recover all of classical thermodynamics 2 Those who rst derive the Boltzmann probabil ity distribution or canonical distribution of a thermalized system then compute some statisti cal quantities which by identi cation with classical equations of thermodynamics lead to the statisti cal entropy formula e g in Tien and Lienhard6 or Sekerka7 Intermediate approaches can be found for instance in Reif8 Chandler 9 or Pathria10 which basically start by de ning arbitrarily the entropy of the microcanonical ensemble Boltzmann entropy then deriving the canon ical distribution from it and nally join the above second category In my opinion the magical side and the arbitrariness of the rst and intermediate approaches can be disturb ing for students as is for many the connection between energy and information i e a number of possibilities of fered to the system For this reason the last category could be my favorite but the main didactic di culty lies in the derivation of the Boltzmann distribution from fun damental assumptions that would be easily stated and acceptable for all The classic way to derive the Boltzmann distribution6 7 is by seeking which distribution is the most probable that is to say which distribution allows the multiplicity of a macrostate to be maximum This classic method presents some didactic di culties Firstly it involves Lagrange multipliers that is usually not introduced in undergraduate courses Secondly it uses also the Stir ling approximation which derivation is far from being straightforward Lastly it is not the multiplicity but its logarithm which is maximized and even if it is the same thing it has a slight taste of arbitrariness and gives the feeling that we know the solution in advance In this note we propose a short derivation of the Boltz mann distribution which avoid these issues and starts from the fundamental postulate of statistical mechanics So very simply the Gibbs formula can be obtained with out su ering via the free energy and the partition func tion I FREE ENERGY Let us consider a system at temperature T denote U the internal energy S the entropy and W the work exchanged with the surroundings For any process that the system can undergo the Clausius inequality writes For processes occurring at constant temperature this can be rewritten as The di erential d U T S introduces a thermodynamical potential11 named free energy or Helmholtz free energy dU T dS W d U T S W F U T S As S F T the later equation can be rewritten as U F T S F T F F T 1 T F 1 T F T 1 T This equation is one of the Gibbs Helmholtz relations 2 3 4 5 II FUNDAMENTAL POSTULATE Em are independent random variables This is possible since N is very large Denote Imagine a large isolated system with constant total energy E divided into a large number N of subparts with number n 1 N The probability for E to have a given value x is E En cst 6 P E x pn i pm x i 9 E En Em w 1 Xi 0 2 8 N Xn 1 Let us de ne a microstate by the multiplet obtained by the energies of subparts microstate E1 E2 EN and denote WN the number of possible microstates the system can adopt A nite resolution for the energy and the existence of a upper boundary allow us to discretize the possible energy levels of subparts into a set of w values En 0 1 w 1 with i i 7 where is the smallest observable energy exchange The system is dynamical and at every time an ele mentary transition can occur from a given microstate to another E1 Ek El EN elementary transitionx y E1 Ek El EN Although elementary transitions are fundamentally de terministic the nite resolution at which the initial mi crostate is known prevent us to predict the nal one The microstate in which the system can be found is thus a random variable Based on the information we have about microstates there is no reason to believe that one is more likely than another The fundamental postulate of statistical me chanics which is a variation of the Laplace s principle of insu cient reason states that at the equilibrium all the WN microstates of an isolated system have equal probability 1 WN III BOLTZMANN DISTRIBUTION The energy level of a given subpart a closed system uctuates due to elementary transitions To simplify the notation let us note the probability P En i that a given subpart n has the energy level i as P En i pn i Consider two given subparts n m su ciently far the one from the other so that they are uncoupled En and Once given n m and E x the rest of the N 2 subparts have a xed energy E x so that it can be viewed as isolated with WN 2 equiprobable microstates in virtue of the fundamental postulate that only depend on E x but not on the peculiar state the subparts n and m adopt In Eq 9 the three events 1 subpart n has energy i 2 subpart m has energy x i and 3 the rest adopts any particular microstate with energy E x are complementary events for any microstate of the whole with probability 1 WN So that one can always write 1 WN pn i pm x i 10 1 WN 2 It follows that in Eq 9 the w terms pn i pm x i of the sum are all the same and equal …