Why are probabilities distribution of thermodynamic variables tend to Gaussian?

[dd331]

The probability distribution for some thermodynamic variable x is given by

$$P = N e^{-A(x)/KT}$$

where A(x) is the availability, which can be replaced by Hemlholtz free energy F, Gibb's free energy G, etc depending on the conditions imposed. N is just some normalization constant. A(x) can be expanded in a taylor series about the equilibrium conditions,

$$A(x) = A(x_{0}) + (x - x_{0})(\frac {\partial A} {\partial x})_{x = x_{0}} + \frac{1} {2} (x - x_{0})^{2} (\frac {\partial^2 A} {\partial x^2})_{x = x_{0}} + ...$$

The second term is 0 since dA/dx = 0 at equilibrium. If we truncate all the other terms, clearly we see that P will be a Gaussian distribution with mean of $$x_{0}$$ and standard deviation of

$$\sqrt {\frac {K T} {(\frac {\partial^2 A} {\partial x^2})_{x = x_{0}}}}$$

What is the justification for truncating this series? This is justified if (x - x0) is small. But why will it be small for big N?

 

[mathman]

I am not familiar with the details of the physics. However such truncation would be based on the assumption |x-x₀| is small.

 

[astrokreng]

The truncation of the Taylor series is justified because in most thermodynamic systems, the fluctuations in the variable x are small compared to its equilibrium value x0. This is because as the system size (represented by N) increases, the fluctuations in x become smaller due to the law of large numbers. As N increases, the probability distribution becomes more sharply peaked around its mean value x0, making the higher order terms in the Taylor series negligible.

Moreover, the Gaussian distribution is a natural result of the central limit theorem, which states that the sum of a large number of independent random variables will tend towards a Gaussian distribution, regardless of the underlying probability distribution of the individual variables. In the case of thermodynamic systems, the random variables can be thought of as the microscopic states of the system, which contribute to the overall macroscopic behavior. As the number of these microscopic states (represented by N) increases, the resulting distribution will tend towards a Gaussian shape.

In summary, the Gaussian distribution for thermodynamic variables is a result of the law of large numbers and the central limit theorem, both of which are applicable in the case of large N. Therefore, it is expected that the probability distribution for thermodynamic variables will tend towards a Gaussian shape.