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Fluctuation, Dissipation, and Non-Boltzmann Energy Distributions

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Emergence of Temperature in Examples and Related Nuisances in Field Theory

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Abstract

After reviewing the field theoretical description of the conventional thermal equilibrium and studying the linear relaxation to this equilibrium, it is interesting to consider problems arising in small systems not satisfying the pre-requisites for the Gibbs–Boltzmann treatment. In this chapter, we discuss general questions of thermal ensembles and the stochastic dynamics picture of a practically unknown environment, and at the end we sketch briefly the way the Keldysh formalism incorporates physical noise into its descriptive framework.

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Notes

  1. 1.

    A notable exception is provided by black hole horizons, where \(S(E)\sim E^2\), \(S^{\prime }(E)\sim E\), and \(S^{\prime \prime }(E) \propto {\mathscr {O}}(1) > 0\), with a seemingly negative specific heat. For classical ideal gases, on the other hand, \(S(E,N)=aN\ln (E/NT_0)\) and \(\partial S/\partial E = aN/E=1/T\), while \(\partial ^2S/\partial E^2 = -aN/E^2=-1/Nc_VT^2\) delivers a positive and constant specific heat at fixed volumes: \(c_N=1/a\).

  2. 2.

    The choice of this range is motivated by the fact that the function \(\sin x/x\) differs from zero mainly in this interval.

  3. 3.

    It has a destabilizing effect for Hamiltonians \(H_\mathrm{s}(P,Q)\) with an energy minimum at \(Q=0\).

  4. 4.

    By analogy with the previously discussed oscillator bath, \(\varphi \sim Q\) and \(\chi \sim q_i\).

  5. 5.

    Pure vacuum contributions, like \(\int S(k)S(p-k)\), are neglected here.

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Authors and Affiliations

  1. H.A.S. Wigner Research Centre for Physics, Budapest, Hungary

    Tamás Sándor Biró

  2. Institute of Physics, Roland Eötvös University, Budapest, Hungary

    Antal Jakovác

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  1. Tamás Sándor Biró
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  2. Antal Jakovác
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Correspondence to Tamás Sándor Biró .

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Biró, T.S., Jakovác, A. (2019). Fluctuation, Dissipation, and Non-Boltzmann Energy Distributions. In: Emergence of Temperature in Examples and Related Nuisances in Field Theory. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-11689-7_5

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