# On the Equilibrium Fluctuations of an Isolated System - Mathematical Physics

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Abstract: Traditionally, it is understood that fluctuations in the equilibriumdistribution are not evident in thermodynamic systems of large \$N\$ the numberof particles in the system \cite{Huang1}. In this paper we examine thevalidity of this perception by investigating whether such fluctuations can inreality depend on temperature.Firstly, we describe fluctuations in the occupation numbers of the energylevels for an isolated system, using previously unknown identities that we havederived for the purpose, which allow us to calculate the moments of theoccupation numbers. Then we compute analytically the probability distributionof these fluctuations. We show that, for every system of fixed and finite \$N\$,fluctuations about the equilibrium distribution do, in fact, depend on thetemperature. Indeed, at higher temperatures the fluctuations can be so largethat the system does not fully converge on the Maxwell-Boltzmann distributionbut actually fluctuates around it. We term this state, where not one macrostatebut a region of macrostates closely fit the underlying distribution, a ``{\itfluctuating equilibrium}. Finally, we speculate on how this finding isapplicable to networks, financial markets, and other thermodynamic-likesystems.

Autor: Kieran Kelly, Przemysław Repetowicz, Seosamh macRéamoinn

Fuente: https://arxiv.org/