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...

 

[eljose79]

Thank you for your interesting question. The approximation of the partition function Z(\beta) in statistical physics and the kinetic theory of gases is a common problem that many scientists have faced. The classical or semiclassical approach you mentioned, where we approximate the partition function as an integral over phase space, is indeed a useful tool for extracting conclusions about the behavior of a system and calculating thermodynamic quantities.

In fact, this semiclassical approach is often used in statistical mechanics to calculate thermodynamic quantities such as specific heat, entropy, and free energy. It is based on the assumption that the phase space distribution of a system is approximately Gaussian, which is a reasonable approximation for many systems at low temperatures. This allows us to use the saddle point method or numerical quadrature methods to evaluate the integral and obtain accurate results for the thermodynamic quantities.

However, it is important to note that this approximation is not always accurate, especially for systems with highly non-linear behavior or at high temperatures. In those cases, the full sum over energy states must be taken into account to accurately describe the system. Also, the semiclassical approach may not be suitable for systems with quantum effects, where the discrete nature of energy levels must be considered.

In conclusion, the semiclassical approximation for the partition function can be a useful tool for calculating thermodynamic quantities in statistical physics and the kinetic theory of gases, but its accuracy depends on the behavior of the energy states and the temperature of the system. It is always important to carefully consider the limitations of any approximation method and to use it in conjunction with other techniques to validate the results. I hope this helps answer your question. Best of luck with your research!

 

[charng]

The difference between a sum and an integral lies in the way they are calculated. A sum is an arithmetic operation that involves adding a sequence of numbers, while an integral is a mathematical operation that involves finding the area under a curve.

In the context of statistical physics and kinetic theory of gases, the sum Z(β) represents the partition function, which is a fundamental quantity used to calculate thermodynamic properties of a system. It is obtained by summing over all possible energy levels of the system, weighted by their respective degeneracies and Boltzmann factors.

However, in some cases, the sum may be difficult to evaluate analytically due to the complexity of the energy levels. In such cases, the integral representation, Z(β)∼∫R2dxdpe−βH(x,p), can be used as an approximation. This integral takes into account the classical Hamiltonian of the system, which is a function of the system's position and momentum. By using this integral, we can extract conclusions about the behavior of the system and calculate thermodynamic quantities such as specific heat, entropy, and free energy.

From the classical or semiclassical point of view, this integral approximation can be accurate enough to extract meaningful conclusions about the system's behavior. However, for low temperatures, T<<1, more sophisticated methods such as the saddle point method or numerical quadrature may be needed to accurately calculate the integral.

In summary, the difference between a sum and an integral lies in their mathematical operations, but in the context of statistical physics and kinetic theory of gases, the integral can be used as an approximation for the sum to calculate thermodynamic properties of a system.