# Why are probabilities distribution of thermodynamic variables tend to Gaussian?

• dd331
In summary, The probability distribution for a thermodynamic variable x is given by P = N e^{-A(x)/KT}, where A(x) is the availability and N is a normalization constant. A(x) can be expanded in a Taylor series around equilibrium conditions and truncating the series results in a Gaussian distribution with mean x0 and standard deviation of \sqrt {\frac {K T} {(\frac {\partial^2 A} {\partial x^2})_{x = x_{0}}}} . This truncation is justified if (x - x0) is small, but the reason for this being small for large N is not specified.
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?

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

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.

## 1. Why are probabilities distribution of thermodynamic variables tend to Gaussian?

The Gaussian distribution, also known as the normal distribution, is a commonly observed distribution in nature. This is because it is a result of the central limit theorem, which states that the sum of many independent random variables will tend towards a Gaussian distribution, regardless of the underlying distribution of the individual variables. Since thermodynamic variables are often the result of many smaller interactions, they tend to follow a Gaussian distribution.

## 2. How does the Gaussian distribution relate to thermodynamic variables?

The Gaussian distribution is a probability distribution that describes the likelihood of a continuous variable taking on a certain value. In thermodynamics, variables such as energy, temperature, and pressure are continuous and can take on a range of values. Therefore, they can be described by a Gaussian distribution to show the probability of them occurring at a certain value.

## 3. Are there any exceptions to the tendency of thermodynamic variables to follow a Gaussian distribution?

While the Gaussian distribution is a commonly observed distribution in nature, there are some cases where it may not accurately describe the distribution of thermodynamic variables. This can occur when there are non-linear or non-Gaussian interactions between the variables, or when the system is in a non-equilibrium state.

## 4. What are the implications of the Gaussian distribution in thermodynamics?

The Gaussian distribution has important implications in thermodynamics, as it allows us to make predictions and calculations about the behavior of thermodynamic variables. For example, it allows us to calculate the probability of a certain value of energy occurring in a system, or to estimate the average value of a variable based on its distribution.

## 5. How does the Gaussian distribution affect our understanding of thermodynamics?

The tendency of thermodynamic variables to follow a Gaussian distribution helps us to better understand and model the behavior of these variables in natural systems. It allows us to make predictions and draw conclusions about the behavior of complex systems, and helps us to better understand the underlying principles of thermodynamics.

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