Exact expressions for the energy current and stress tensor operators in many-particle systems

For some purposes in statistical physics, such as, for example, the calculation of various transport coefficients, it is necessary to have expressions for the energy current operatorS and stress tensor operatorT _lm . In this work it is shown that by using a simple identity, exact expressions forS andT _lm which satisfy the conservation laws for the energy density? and momentum densityP, respectively, exist.S andT _lm can each be written as a sum of two parts,S=S ^(A) +S ^(B) T _lm =T _lm ^(A) +T _lm ^(B) . The “A” part is the ordinary convective or kinetic part while the “B” part is shown to be expressible as a gradient and hence its homogeneous component vanishes identically. The expressions are compared with approximate forms found in the literature. The operators are Fourier analyzed and written in terms of the field operators in the second quantization formalism.