Difference between a sum and an integral

eljose

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

$$Z(\beta)= \sum_{n} g(n)e^{-\beta E_n }$$

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

$$Z(\beta)\sim \int_{R^2}dxdpe^{-\beta H(x,p)}$$

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-

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

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

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"Difference between a sum and an integral"

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