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Classical Virial Theorem (canonical ensemble derivation)

Again, let \(x_i\) and \(x_j\) be specific components of the phase space vector \(x = (p_1,\cdots ,p_{3N},q_1,\cdots,q_{3N})\). Consider the canonical average

\[ \langle x_i \frac {\partial H}{\partial x_j} \rangle \]


given by
 

\( \langle x_i \frac {\partial H}{\partial x_j} \rangle \)

$\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\frac {1}{Q} C_N \int dx x_i  \frac {\partial H}{\partial x_j}e^{-\beta H(x)}\)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\frac {1}{Q} C_N \int dx x_i \left(- \frac {1}{\beta} \frac {\partial}{\partial x_j} \right ) e^{-\beta H(x)}\)

 


But

\( x_i \frac {\partial}{\partial x_j}e^{-\beta H(x)}\)

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\(\frac {\partial}{\partial x_j} \left ( x_i e^{-\beta H(x)} \right ) - e^{-\beta H(x)} \frac {\partial x_i}{\partial x_j}\)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\frac {\partial}{\partial x_j} \left ( x_i e^{-\beta H(x)} \right ) - \delta_{ij}e^{-\beta H(x)}\)

 


Thus,

\( \langle x_i \frac {\partial H}{\partial x_j} \rangle \)

$\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

 \( - \frac {1}{\beta Q} C_N \int dx \frac {\partial}{\partial x_j} \left ( x_i e^{- \beta H (x)} \right ) + \frac {1}{\beta Q} \delta_{ij} C_N \int dx e^{-\beta H(x)}\)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\( - \frac {1}{\beta Q} C_N \int dx'  \int dx_j \frac {\partial}{\partial x_j} \left ( x_i e^{-\beta H(x)}\right )+ kT\delta_{ij}\)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\( \int dx' \left.x_i e^{-\beta H(x)} \right \vert _{x_j=-\infty}^{\infty}+ kT\delta_{ij}\)

 

Several cases exist for the surface term \(x_i  exp  (-\beta H (x)) \):

  1. \(x_i = p_i \)a momentum variable. Then, since \(H \sim p^2_i,   exp  (-\beta H)\) evaluated at \(p_i = \pm \infty \) clearly vanishes.
  2. \(x_i = q_i \)and \(U \rightarrow \infty \) as \(q_i \rightarrow \pm \infty \), thus representing a bound system. Then, \(exp  (- \beta H ) \) also vanishes at \(q_i = \pm \infty \).
  3.  \(x_i = q_i \)and \(U \rightarrow 0 \) as \(q_i \rightarrow \pm \infty \), representing an unbound system. Then the exponential tends to 1 both at \(q_i = \pm \infty \), hence the surface term vanishes.
  4. \(x_i = q_i \)and the system is periodic, as in a solid. Then, the system will be represented by some supercell to which periodic boundary conditions can be applied, and the coordinates will take on the same value at the boundaries. Thus, \(H \) and \(exp  (- \beta H) \) will take on the same value at the boundaries and the surface term will vanish.
  5. \(x_i = q_i \) and the particles experience elastic collisions with the walls of the container. Then there is an infinite potential at the walls so that \(U \rightarrow \infty \) at the boundary and \( exp (- \beta H ) \rightarrow 0 \) at the boundary.

Thus, we have the result

\[\langle x_i \frac {\partial H}{\partial x_j} \rangle = kT\delta_{ij}\]

 

The above cases cover many but not all situations, in particular, the case of a system confined within a volume \(V\) with reflecting boundaries. Then, surface contributions actually give rise to an observable pressure (to be discussed in more detail in the next lecture).