Difference between a sum and an integral (1 Viewer)

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Let,s suppose we have in statistical physics /Kinetic theory of gases the "sum"...

[tex] Z(\beta)= \sum_{n} g(n)e^{-\beta E_n } [/tex]

Of course depending on the behavior of {E_n} the sum will be difficult to evaluate..my question is if from the classical or semiclassical point of view the approximation

[tex] Z(\beta)\sim \int_{R^2}dxdpe^{-\beta H(x,p)} [/tex]

Where H is the classical Hamiltonian of the system..will be accurate enough to extract conclussions about the behavior of the systme and calculate Thermodynamical quantities (Specific Heat Cp,Cv,F,G,H,S)..thanks-:rolleyes: :rolleyes:

Where the Hamiltonian is usually of the form [tex] H=p^2 +V(x) [/tex] so in the end we deal with integrals of the form:

[tex] \int_{-\infty}^{\infty}dxe^{-\beta V(x)} [/tex] so for T<<1 we could perform a "Saddle point method" or simpli Numerical Cuadrature methods...
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