Virial theorem of Clausius

That this is indeed the differential form of the customary virial theorem is readily seen by multiplying Eq. (26) throughout by x and then integrating over all x from —oo to +00. Some elementary integrations by parts recovers the usual (integral) virial theorem of Clausius, in, of course, now fully quantum-mechanical form [54]. 

The pressure is usually calculated in a computer simulation via the virial theorem of Clausius. The virial is defined as the expectation value of the sum of the products of the coordinates of the particles and the forces acting on them. This is usually written W = Pxi where X is a coordinate (e.g. the x or y coordinate of an atom) and p. is the first derivative of the momentum along that coordinate pi is the force, by Newton s second law). The virial theorem states that the virial is equal to —3Nk T. 

The virial theorem of Clausius estabhshes the ratio of the average kinetic energy K to the average potential energy V of a mechanical system consisting of i particles [28, 33—36). The virial of the force F is equal to the total average kinetic energy K ... 

TUs is often called the viiial equation for the pressure since it can also be derived firom the virial theorem of classical mechanics of Clausius. The product ru (r) is the virial of the pair potential u(r). 

In the original derivation of the classical virial theorem given by Clausius, an expression corresponding to eqn (5.30) is also obtained. In the classical case one argues that the time average of d(f p)/dt vanishes over a sufficiently long period of time or that the motion is periodic to obtain the equivalent of eqn (5.31). ... 

The virial theorem was first stated by Clausius in 1870 for the expectation value of the product between the position of each particle and the force acting on it. Indeed, substituting in Eq. 8.15 for the position of a particle, and using Hamilton s equation of motion, yields... 

---