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I'm following the derivation of the equipartition theorem for the microcanonical ensemble, as I have seen it on wikipedia. In the end, it derives: $$ \langle x_m \frac{\partial H}{\partial x_n} \rangle = \delta_{nm} \frac{1}{\frac{\frac{\partial \Omega}{\partial E}}{\Omega}} $$ And then they use $S = \log \Omega$ to arrive at the temperature. What bugs me about this is that $\Omega$ is not the partition sum of the microcanonical ensemble here:

$$ \Omega = \int_{H<E} d\Gamma $$ While $$ Z_{\text{microcanonical}} = \int_{H=E} d\Gamma \approx \frac{1}{\Delta E} \int_{E < H < E + \Delta E} $$

Here $\Gamma$ denotes a microstate (in statistical mechanics, that would be an element of phase space). I have read that for large N, most of the volume lies in the surface of a hypersphere with dimension N, and because of that this approximation of $\Omega = Z$ can be made, but does that also mean that the equipartition theorem for the microcanonical ensemble is only valid for large N as well?

To me that's shocking, as the equipartition theorem is the way of obtaining microscopic temperatures from systems (for example when you simulate a system with 100 particles and then calculate a temperature for each timestep). Can we somehow safe the derivation?

Edit: As I have read now in thisquestion, there is a different entropy used in this formula that is also called "Gibbs Entropy". This entropy strangely yields the correct temperature in the equipartition theorem.

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    $\begingroup$ What is the meaning of $\int H$ and $E d\Gamma$? $\endgroup$ Jan 17, 2022 at 2:31
  • $\begingroup$ @GiorgioP I corrected $\endgroup$ Jan 17, 2022 at 3:14

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I have met the same problem reading that. I did some research and found this paper helpful: https://doi.org/10.1063/1.2889939

The authors of this paper argue that for microcanonical ensemble, it is the Boltzmann-Planck definition $S=k\ln\omega$ that should be used, where $\omega$ is the number of microstates with total energy being held strictly fixed at $E$. The equally division of kinetic energy is still valid:

$$\langle\mathbf{p}_i\cdot\nabla_{\mathbf{p}_j}H\rangle = \delta_{ij}2\frac{\langle K\rangle}{N} = \delta_{ij}\frac{2}{N}\langle\sum_{i=1}^N\frac{p_i^2}{2m_i}\rangle$$

But because of the use of $\omega$ instead of $\Omega$, now $\langle K \rangle$ has no direct relation to the temperature. The temperature is related to the kinetic energy by:

$$\left(\frac{3N}{2}-1\right)kT=\left[\langle K^{-1}\rangle\right]^{-1}$$

This implies that if we want to calculate the temperature of a microcanonical ensemble, we should calculate the ensemble average of the inverse of the kinetic energy.

The result above can be derived as follows. We assume that the Hamiltonian can be expressed as

$$H = \sum_{i=1}^N \frac{\mathbf{p}_i^2}{2m_i} + U(\mathbf{r}_1\ldots,\mathbf{r}_N)$$

The number of microstates in a microcanonical ensemble is

$$\omega(N,V,E) = \frac{\Delta E}{g} \int\delta(E-H) d^{3N}rd^{3N}p$$

So the temperature can be calculated as

$$\frac{1}{kT} = \frac{\partial\ln\omega}{\partial E} = \frac{1}{\omega} \frac{\Delta E}{g} \int\delta^\prime(E-H) d^{3N}r d^{3N}p$$

To evaluate that, we can perform the Laplace transform on it ($E$ to $s$):

\begin{aligned} \mathcal{L}\left[\frac{\partial\ln\omega}{\partial E}\right] &= \frac{1}{\omega} \frac{\Delta E}{g} \int\int\delta^\prime(E-H)e^{-sE}dE\ d^{3N}r d^{3N}p \\&= \frac{1}{\omega} \frac{\Delta E}{g} \int se^{-sH}\ d^{3N}r d^{3N}p \\&= \frac{1}{\omega} \frac{\Delta E}{g} \int s\exp[ -s (\sum_{i=1}^N \frac{\mathbf{p}_i^2}{2m_i} + U)] d^{3N}rd^{3N}p \\&= \frac{1}{\omega} \frac{\Delta E}{g}(2\pi)^{3N/2} \prod_{i=1}^Nm_i^{3/2} s^{-(3N/2-1)} \int e^{-sU}d^{3N}r \end{aligned}

By inverse Laplace transform we can get

$$ \frac{1}{kT} = \frac{1}{\omega} \frac{\Delta E}{g} (2\pi)^{3N/2} \prod_{i=1}^Nm_i^{3/2} \int \frac{(E-U)^{3N/2-2}}{\Gamma(3N/2-1)} \Theta(E-U) d^{3N}r$$

where $\Theta$ is the Heaviside step function. On the other hand, the ensemble average of the inverse of kinetic energy is

$$\langle K^{-1}\rangle = \langle (E-U)^{-1}\rangle = \frac{1}{\omega} \frac{\Delta E}{g} \int \frac{\delta(E-H)}{E-U} d^{3N}pd^{3N}r$$

The Laplace transform of this is

\begin{aligned} \mathcal{L}\left[\langle K^{-1}\rangle\right] &= \frac{1}{\omega} \frac{\Delta E}{g} \int \frac{e^{-sH}}{H-U}d^{3N}pd^{3N}r \\&= \frac{1}{\omega} \frac{\Delta E}{g} \int \frac{ \exp(-s\sum_{i=1}^N \frac{p_i^2}{2m_i}) }{ \sum_{i=1}^N \frac{p_i^2}{2m_i} } d^{3N}p \int \exp(-sU) d^{3N}r \end{aligned}

To calculate the integral over $p$, we can make a coordinate transform $p^\prime_i = p_i/\sqrt{2m_i}$, and let $\xi^2 = \sum {p^\prime}^2$:

\begin{aligned} \mathcal{L}\left[\langle K^{-1}\rangle\right] &= \frac{1}{\omega} \frac{\Delta E}{g} 2^{3N/2} \prod_{i=1}^Nm_i^{3/2} \int \frac{e^{-s\xi^2}}{\xi^2} d^{3N}p^\prime \int e^{-sU} d^{3N}r \\&=\frac{1}{\omega} \frac{\Delta E}{g} 2^{3N/2} \prod_{i=1}^Nm_i^{3/2} \int_0^\infty \frac{e^{-s\xi^2}}{\xi^2} A(\xi)d\xi \int e^{-sU} d^{3N}r \end{aligned}

where $A(\xi)$ is the surface of a 3N-dimension sphere with radius $\xi$ and is equal to

$$ A(\xi) = \frac{2\pi^{3N/2}}{\Gamma(3N/2)}\xi^{3N-1}$$

Substitution of this into the integration above yields

$$\mathcal{L}\left[\langle K^{-1}\rangle\right] = \frac{1}{\omega} \frac{\Delta E}{g} (2\pi)^{3N/2} \prod_{i=1}^N m_i^{3/2} \frac{1}{3N/2-1} \int \frac{e^{sU}}{s^{3N/2-1}} d^{3N}r$$

By inverse Laplace transform, we can finally get

$$\langle K^{-1} \rangle = \frac{1}{\omega} \frac{\Delta E}{g} (2\pi)^{3N/2} \prod_{i=1}^N m_i^{3/2} \frac{1}{3N/2-1} \int \frac{(E-U)^{3N/2-2}}{\Gamma(3N/2-1)} \Theta(E-U) d^{3N}r$$

By comparing this with the result of $1/kT$, we can easily find

$$\left(\frac{3N}{2}-1\right)kT=\left[\langle K^{-1}\rangle\right]^{-1}$$

So using different definitions of entropy yields different results. The authors elaborated on why they believe $S=k\ln\omega$ is the right one to use. They calculated the distribution of speeds using both $\omega$ and $\Omega$ and demonstrated that only the former is consistent with their MD simulation of 10 hard-sphere particles. Since the entropy can also be seen as an ensemble average ($S=-k\langle \ln\rho\rangle$, $\rho$ is the probability density of a microstate), it is reasonable to do it in the phase space $\omega$ too. Also, they found that $\langle K\rangle_\Omega=\sum\langle p_i^2\rangle_\Omega / 2m_i \not= \langle E-U\rangle_\Omega$ and this result is considered as not physical. Those who approve the use of $S=k\ln\Omega$ often cite the result that $\Omega$ is adiabatically invariant, but this reason is insufficient in the authors' opinion.

The authors also derived the result for $NVE\mathbf{PG}$ ensemble in which total momentum $\mathbf{P}$ and $\mathbf{G}=t\mathbf{P} - M_{total}r_c$ are also constant. This is important for molecular dynamics simulation because most of the time we do the "$NVE$ ensemble MD", we only simulate once and then calculate the time average. If there is no external force, what we get is actually a $NVE\mathbf{PG}$ ensemble. The results are slightly different:

$$\langle\frac{\mathbf{p}_i^2}{2m_i}\rangle = \frac{M_{totle}-m_i}{M_{total}(N-1)}\langle E-U\rangle + \frac{Nm_i-M_{total}}{2(N-1)M_{total}^2}\mathbf{P}^2$$

$$\left(\frac{3(N-1)}{2}-1\right)kT=\left[\left\langle \left(E-U-\frac{\mathbf{P}^2}{2M_{total}}\right) ^ {-1} \right\rangle\right]^{-1}$$

In the thermodynamic limit where $N$ is large, the result of the equipartition theorem and the results derived in this paper for $NVE$ and $NVE\mathbf{PG}$ ensemble become equivalent. So in that case it's still safe to use the equipartition theorem to calculate the temperature in $NVE$ MD simulations.

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  • $\begingroup$ Thank you for the paper. The including discussion is extremely enlightening. . $\endgroup$ Jan 31, 2022 at 0:19

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