<P> Since the system can be in any energy eigenstate within an interval of δ E (\ displaystyle \ delta E), we define the generalized force for the system as the expectation value of the above expression: </P> <Dl> <Dd> X = − ⟨ d E r d x ⟩ (\ displaystyle X = - \ left \ langle (\ frac (dE_ (r)) (dx)) \ right \ rangle \,) </Dd> </Dl> <Dd> X = − ⟨ d E r d x ⟩ (\ displaystyle X = - \ left \ langle (\ frac (dE_ (r)) (dx)) \ right \ rangle \,) </Dd> <P> To evaluate the average, we partition the Ω (E) (\ displaystyle \ Omega \ left (E \ right)) energy eigenstates by counting how many of them have a value for d E r d x (\ displaystyle (\ frac (dE_ (r)) (dx))) within a range between Y (\ displaystyle Y) and Y + δ Y (\ displaystyle Y+ \ delta Y). Calling this number Ω Y (E) (\ displaystyle \ Omega _ (Y) \ left (E \ right)), we have: </P>

The first law of thermodynamics is a re – statement of the law of