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Necessary Background of Statistical Physics (2)

Necessary Background of Statistical Physics (2). Temperature. N c = 3 when the velocity of the center of mass is fixed to zero. Kinetic energy:. Pressure. Definition:. Working expression:. Chemical potential m = (  F/  N) T,V Widom method (test particle method):

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Necessary Background of Statistical Physics (2)

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  1. Necessary Background of Statistical Physics (2)

  2. Temperature Nc = 3 when the velocity of the center of mass is fixed to zero. Kinetic energy:

  3. Pressure Definition: Working expression:

  4. Chemical potential m = (F/N)T,V Widom method (test particle method): m = FN+1 - FN = -kTln(ZN+1/ZN) = kTln3 - kTln(QN+1/QN)  = (2pb2/m)1/2 - thermal wave length where is the interaction potential between the N+1 th particle with with all the others, mex = m - mid = -kT ln(<exp(-bf)>N) mid = kT ln(r3) - chemical potential of the ideal gas Drawback: break down at high densities!

  5. Properties which can be determined from fluctuations Important remark: Fluctuations are ensemble dependent! But averages are not. e.g., <ΔH(pN,qN)2>=0 in the microcanonical ensemble but is non zero in the canonical ensemble. Heat capacity kT2CV = <ΔH(pN,qN)2>NVT = <(H(pN,qN) - <H(pN,qN) >NVT)2 >NVT = <H(pN,qN)2>NVT - <H(pN,qN)>NVT2 Remark: In general, the numerical precision on fluctuations is poorer than that for the averages. <ΔH(pN,qN)2>NVT = <ΔU2>NVT + <ΔK2>NVT <ΔK2>NVT = 3N(kT)2/2

  6. Isothermal compressibility rkTχT = <DN2>mVT/<N>mVT kT(<N>mVT /m)V,T = <DN2>mVT

  7. Transport properties Self-diffusion coefficient Einstein formula: <|r(t) - r(0)|2>  6Dt t   Expression in terms of velocity auto-correlation function (VACF): Diffusion equation: Solution with the initial condition, r(r, t=0) = d(r - r0):

  8. <|r(t) - r(0)|2> 1 t 0 t Mean square displacement Velocity auto-correlation function Remark: The method of mean square displacement provides a better numerical precision in the calculation of D.

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