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Energy fluctuations in the canonical ensemble

  • Page ID
    5231
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    In the canonical ensemble, the total energy is not conserved. ( \(H (x) \ne \text {const} \) ). What are the fluctuations in the energy? The energy fluctuations are given by the root mean square deviation of the Hamiltonian from its average \(\langle H \rangle \):

    \[ \Delta E = \sqrt{\langle\left(H-\langle H\rangle\right)^2\rangle} =\sqrt{\langle H^2 \rangle - \langle H \rangle^2}\]

    \(\langle H \rangle \)

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    \( - \frac {\partial}{\partial \beta} \ln Q (N,V,T) \)

    \(\langle H^2 \rangle \)

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    \(\frac {1}{Q} C_N \int dx H^2 (x) e^{- \beta H (x)} \) }}

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    \( \frac{1}{Q} C_N \int dx \frac{\partial^2}{\partial \beta^2}e^{-\beta H(x)}\)

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    \( \frac{1}{Q} \frac {\partial^2}{\partial \beta^2}Q\)

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    \(\frac{\partial^2}{\partial \beta^2}\ln Q + \frac {1}{Q^2} \left( \frac {\partial Q}{\partial \beta}\right)^2\)

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    \(\frac{\partial^2}{\partial \beta^2}\ln Q +\left[\frac{1}{Q} \frac{\partial Q}{\partial \beta}\right]^2\)

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    \(\frac{\partial^2}{\partial \beta^2}\ln Q + \left[ \frac {\partial}{\partial \beta}\ln Q\right]^2\)


    Therefore

    \[\langle H^2 \rangle - \langle H\rangle^2 =\frac{\partial^2}{\partial \beta^2}\ln Q\]


    But

    \[ \frac{\partial^2}{\partial \beta^2}\ln Q = kT^2 C_V\]


    Thus,

    \[\Delta E = \sqrt{kT^2 C_V}\]


    Therefore, the relative energy fluctuation \(\frac {\Delta E}{E} \) is given by

    \[\frac{\Delta E}{E} = \frac{\sqrt{kT^2 C_V}}{E}\]


    Now consider what happens when the system is taken to be very large. In fact, we will define a formal limit called the thermodynamic limit, in which \(N\longrightarrow\infty\) and \(V\longrightarrow\infty\) such that \(\frac {N}{V} \) remains constant.

    Since \(C_V\) and \(E\) are both extensive variables, \(C_V\sim N\) and \(E \sim N\),

    \[\frac {\Delta E}{E} \sim \frac{1}{\sqrt{N}} \longrightarrow 0\;\;\;{as}\;\;\;N\rightarrow \infty\]


    But \(\frac {\Delta E}{E} \) would be exactly 0 in the microcanonical ensemble. Thus, in the thermodynamic limit, the canonical and microcanonical ensembles are equivalent, since the energy fluctuations become vanishingly small.

    Contributors and Attributions

    Mark Tuckerman (New York University)


    Energy fluctuations in the canonical ensemble is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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