In the NVT ensemble (constant Number of particles, Volume, and Temperature), we are considering a system with a fixed number of particles, enclosed in a fixed volume, and in thermal equilibrium with a heat reservoir at a fixed temperature.

When we talk about the distribution of energy in this ensemble, we are referring to how the energy of the particles is distributed among the different possible quantum states that the system can occupy.

In many physical systems, when the number of particles becomes very large (the thermodynamic limit), and the interactions between the particles are not extremely strong, the distribution of energy tends to become Gaussian. A Gaussian distribution is also called a normal distribution or bell curve. It is characterized by a symmetric shape, with most of the data concentrated around the mean value.

The Gaussian distribution means that most of the particles have energies close to the average energy of the system, and there are very few particles with extremely high or extremely low energies. This is why we often observe that macroscopic systems, like gases or solids, show a nearly Gaussian distribution of energy in the NVT ensemble.

As for the statement that "fluctuations of energy in the thermodynamic limit are zero," it needs a slight clarification. In the thermodynamic limit, which means a very large number of particles, the relative fluctuations of energy become very small compared to the average energy. However, this does not mean that fluctuations are exactly zero. Quantum mechanics and statistical mechanics predict that even in the thermodynamic limit, there will still be some tiny fluctuations around the average energy due to the probabilistic nature of quantum systems.

So, while fluctuations become negligible in the thermodynamic limit, they are not exactly zero. It is a consequence of the statistical behavior of a large number of particles in the system, and it is one of the key principles of statistical mechanics that allows us to understand the macroscopic properties of matter based on the microscopic behavior of individual particles.

The probability that the system in NVT ensemble has energy lying between E to E+dE is proportional to e^-𝛽E wn(E). Here wn(E) represents density of state around the energy value E and e^-𝛽E is Boltzmann factor. Here wn(E) is dimensionless. Here wn(E) plays role of weight factor for the energy level E. The actual probability is determined by weight factor wn(E) and as well as factor e^- 𝛽E of level.

On normalizing, we get

where denominator is total probability. Hence denominator can be considered as the partition function of system.

Now,

where S(E) = KB ln wn(E) is entropy in NVE ensemble. Now, let's consider integrated in equation 3

e^-𝛽E + 𝛽TS is function of E and has maxima and let it be

The integrated equation 3 has maxima for energy equal to internal energy N

Again

T>0 and Cv >0 for a physical condition, equation 4 is satisfied. Let's expand TS(E) - E around

Energy around fluctuation of energy is

Relative fluctuation in energy is given by

Cv and U are extensive quantity

ΔE/ U ∼ √N/ N ∼ 1/ √N -> 0 as N-> ∞. This shows that in thermodynamic limit, the relative fluctuation of energy tends to zero.

For NVE, ΔE = 0 in thermodynamic limit. Hence, we conclude canonical ensemble and Grand canonical ensemble are said to be equivalent in thermodynamics limit. The plot of P(E) and E shows that Gaussian distribution of energy ion NVT ensemble centered at E =U with width equal to

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