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Temperature Fluctuations for a System in Contact with a Heat Bath

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Abstract

The problem of how to define and compute temperature fluctuations for a small system in contact with a heat bath is an old one and originates from Einstein’s theory of Brownian motion. Only for a small enough system does their relative size allow a straightforward experimental verification. Here we focus on a mesoscopic system in contact with a heat bath at temperature T ∘ and provide a self-consistent argument showing as to why, and in what sense, the observable standard deviation of temperature from T ∘ equals \(\sqrt{k_{\rm B}/C}T_{\circ}\) where C is the mesoscopic system’s heat capacity. Our argument is based on ergodic decomposition, a simple fact that holds for a system in thermodynamic equilibrium and away from phase-transition points. In this way we close a line of argument opened by de Haas-Lorentz a century ago.

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Notes

  1. Formally, Δ(1/T)=(ΔT)/T 2 so that with \(\Delta E = \sqrt{k_{\rm B} C} T\) we end up with (1).

  2. See in particular pp. 75–81 of ter Haar [27] and read Kramers’ original paper [25] and/or Dresden’s 1988 comments [26] to catch how well Kramers realized the importance of taking the thermodynamic limit.

  3. Whereas even classical Heisenberg spins give rise to a “natural” dynamics, Ising spins do not. One must give them one such as Monte Carlo or Metropolis. The dynamics of the infinite classical Heisenberg model has been shown to exist by [29], and in full detail in [30].

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Acknowledgements

A.L. thanks the Humboldt Foundation for a Humboldt Award allowing his stay at the Technische Universität München. J.L.v.H. gratefully acknowledges the hospitality of the University of Ottawa, where this paper was finished. Both thank Aernout van Enter for constructive criticism.

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  1. Physik Department T35, Technische Universität München, 85747, Garching bei München, Germany

    J. Leo van Hemmen

  2. Department of Physics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada

    André Longtin

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  1. J. Leo van Hemmen
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Correspondence to André Longtin.

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Leo van Hemmen, J., Longtin, A. Temperature Fluctuations for a System in Contact with a Heat Bath. J Stat Phys 153, 1132–1142 (2013). https://doi.org/10.1007/s10955-013-0867-9

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