Virial theorem
In statistical mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by a conservative force (where the work done is independent of path) with that of the total potential energy of the system. Mathematically, the theorem states

  
    
      
        
          ⟨
          T
          ⟩
        
        =
        −
        
          
            1
            2
          
        
        
        
          ∑
          
            k
            =
            1
          
          
            N
          
        
        
          
            ⟨
          
        
        
          
            F
          
          
            k
          
        
        ⋅
        
          
            r
          
          
            k
          
        
        
          
            ⟩
          
        
      
    
    {\displaystyle \left\langle T\right\rangle =-{\frac {1}{2}}\,\sum _{k=1}^{N}{\bigl \langle }\mathbf {F} _{k}\cdot \mathbf {r} _{k}{\bigr \rangle }}
  

where T is the total kinetic energy of the N particles, Fk represents the force on the kth particle, which is located at position rk, and angle brackets represent the average over time of the enclosed quantity. The word virial for the right-hand side of the equation derives from vis, the Latin word for "force" or "energy", and was given its technical definition by Rudolf Clausius in 1870. The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in statistical mechanics; this average total kinetic energy is related to the temperature of the system by the equipartition theorem. However, the virial theorem does not depend on the notion of temperature and holds even for systems that are not in thermal equilibrium.