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    Cmat Term Paper: 1 Phase Space and Ensembles

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    iamrohan
    The document discusses phase space and ensembles in statistical mechanics. It introduces the microcanonical, canonical, and grand canonical ensembles. It then discusses the harmonic oscillator model, including the equations of motion for an undamped and damped oscillator. It derives expressions for the response function and dissipation rate of a damped oscillator. Key relationships presented include the partition functions relating to thermodynamic potentials and expressions for average energy, number of particles, and fluctuations in these quantities.

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    Cmat Term Paper: 1 Phase Space and Ensembles

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    iamrohan
    38 views5 pages
    The document discusses phase space and ensembles in statistical mechanics. It introduces the microcanonical, canonical, and grand canonical ensembles. It then discusses the harmonic oscillator model, including the equations of motion for an undamped and damped oscillator. It derives expressions for the response function and dissipation rate of a damped oscillator. Key relationships presented include the partition functions relating to thermodynamic potentials and expressions for average energy, number of particles, and fluctuations in these quantities.

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    The document discusses phase space and ensembles in statistical mechanics. It introduces the microcanonical, canonical, and grand canonical ensembles. It then discusses the harmonic oscillator model, including the equations of motion for an undamped and damped oscillator. It derives expressions for the response function and dissipation rate of a damped oscillator. Key relationships presented include the partition functions relating to thermodynamic potentials and expressions for average energy, number of particles, and fluctuations in these quantities.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    38 views5 pages

    Cmat Term Paper: 1 Phase Space and Ensembles

    Uploaded by

    iamrohan
    The document discusses phase space and ensembles in statistical mechanics. It introduces the microcanonical, canonical, and grand canonical ensembles. It then discusses the harmonic oscillator model, including the equations of motion for an undamped and damped oscillator. It derives expressions for the response function and dissipation rate of a damped oscillator. Key relationships presented include the partition functions relating to thermodynamic potentials and expressions for average energy, number of particles, and fluctuations in these quantities.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 5

    CMAT

    Term Paper
    Rohan Sinha

    1 Phase Space and ensembles


    Let’s first see the probability of occurence of a particle at a point for various
    systems:

    1. Microcanonical Ensemble

    δ[E − H(R, V )]
    Pmicro (R) =
    ω(E, V )
    , where ω is the density of states chosen such that the sum of P over all
    the states is unity (1). The micro canonical ensemble is not the most useful
    for doing the calculations, because it involves difficult sums over states the
    constrained energies. The other two are far more useful.

    2. In the canonical ensemble, the density matrix is e−βH . Here β = T −1 as


    we will use the units in which kB is equal to one. Hence,
    1
    Pc = e−βH
    ZN (T, V )

    where ZN (T, V ) = T re−βH is the partition function

    3. Similarly, for the Grand Canonical Ensemble,


    1
    Pgc = e−β
    Ξ(T, µV )

    where as earlier, Ξ = T re−β(H−µN )

    The average energy is the the average of the Hamiltonian over the ensemble.

    hEi ≡ hHi = T rP H

    1
    where P is either Pgc or Pc . hEi can be obtained in either ensemble by differenti-
    ation wrt β:
    ∂lnZN
    hEi = −
    ∂β N,V
    or
    ∂lnΞ
    hEi = −
    ∂β βµ,V
    On further calculations, we get:

    ∂ 2 lnZN ∂hEi
    h(δE)2 i ≡ hE 2 i − hEi2 = 2
    =−
    ∂β ∂β
    and a similar expression in terms of Ξ.
    in the case of grand canonical, the number of particles is not fixed and we have:
    1
    hN i = T re−β[H−µN ] (N − hN i)2
    Ξ
    ∂lnΞ ∂ 2 lnΞ ∂hN i
    This gives hN i = and h(δN )2 i = 2
    = . Since both E and N are
    ∂βµ ∂(βµ) ∂βµ
    extensive quantities which grow with the size of the samplem in large systems the
    distinction between the fixed energy of the microcanonical and average energy of
    the canonical and grand canonical ensembles becomes unimportant.
    The partition functions are related to thermodynamic potential as follows:

    F (T, N, V ) = −T ln ZN (T, V )

    A(T, µ, V ) = −T ln Ξ(T, µ, V )
    These relations are obtained by recognizing that ω(E)e−βE peaks near E = hEi.
    From here, we also get:

    and

    2
    2 The Harmonic oscillator
    2.1 The undamped Oscillator
    The Hamiltonian for an undamped oscillator of mass m and spring constant k is:

    the equations of motion for position and momentum can be calculated by taking
    their Poisson bracket with the Hamiltonian. we assume that both are prop to eιωt
    and on solving, we get:

    2.2 damped Oscillator


    We can find the mode structure of the damped oscillator with the equation:

    which in turn gives us:

    On further calculations, we can obtain the approx eqn of motion as:

    2.3 Response Function


    the resfonse function to an external force f can be calculated as:

    3
    On plotting χ(ω) and χ0 (ω) we obtain the following graph for ω when it is real:

    2.4 Dissipation
    The rate at which external force does work is:

    Since in steady state, both f (t) and ẋ(t) are periodic functions the average power
    dissipated is:

    which can be calculated as:

    Thus, we see that the rate of dissipation is proportional to

    4
    References
    [1] P.M Chaikin & T.C.Lubensky Principles of Condensde Matter Physics, Cam-
    bridge University Press

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