Nonequilibrium Statistical Operator DOI: http://dx.doi.org/10.5772/intechopen.84707

commutes with the Hamiltonian. We conclude that ρeq depends only on constants of motion C<sup>n</sup> that commute with H. But, the von Neumann equation is not sufficient to determine how ρeq depends on constants of motion Cn. We need a new

Equilibrium statistical mechanics is based on the following principle to determine the statistical operator ρeq: consider the functional (information entropy)

(self-consistency conditions). Respecting these conditions, we vary ρ and determine the maximum of the information entropy for the optimal distribution ρeq so that δSinf½ ρeq� ¼ 0. As it is well-known, the method of Lagrange multipliers can be used to account for the self-consistency conditions (11). The corresponding

¼ �kBTr ρeqln ρeq

is the equilibrium entropy of the system at given constraints Ch i<sup>n</sup> and k<sup>B</sup> is the Boltzmann constant. The solution of this variational principle leads to the Gibbs

As an example, we consider an open system which is in thermal contact and particle exchange with reservoirs. The corresponding equilibrium statistical operator has to obey the given constraints: normalization Trf gρ ¼ 1, thermal contact with the bath so that Trf g ρH ¼ U (internal energy), particle exchange with a reservoir so that for the particle number operator N<sup>c</sup> of species c, the average is given by Trf g ρN<sup>c</sup> ¼ ncΩ, where Ω denotes the volume of the system (we do not use V to avoid confusion with the potential), and nc is the particle density of species c. Looking for the maximum of the information entropy functional with these

> �<sup>β</sup> <sup>H</sup>�∑<sup>c</sup> ð Þ <sup>μ</sup>cN<sup>c</sup> Tre�<sup>β</sup> <sup>H</sup>�∑<sup>c</sup> ð Þ <sup>μ</sup>cN<sup>c</sup>

The normalization is explicitly accounted for by the denominator (partition function). The second condition means that the energy of a system in heat contact with a thermostat fluctuates around an averaged value Hh i¼ U ¼ uΩ with the given density of internal energy u. This condition is taken into account by the Lagrange multiplier β that must be related to the temperature, a more detailed discussion leads to β ¼ 1=ð Þ kBT . Similarly, the contact with the particle reservoir fixes the particle density nc, introduced by the Lagrange multiplier μc, which has the

Within the variational approach, the Lagrange parameters β, μ<sup>c</sup> have to be eliminated. This leads to the equations of state (h i … eq ¼ Trfρeq…g) which relate, e.g.,

h i H eq ¼ U Ω; β; μ<sup>c</sup> ð Þ, h i N<sup>c</sup> eq ¼ Ωnc T; μ<sup>c</sup> ð Þ: (14)

n o

for arbitrary ρ that is consistent with the given conditions Trf g¼ ρ 1

Sinf½ �¼� ρ Trf g ρln ρ (10)

Trf g ρC<sup>n</sup> ¼ h i C<sup>n</sup> (11)

(12)

: (13)

additional principle, not included in Hamiltonian dynamics.

Seq ρeq h i

ensembles for thermodynamic equilibrium, see also Section 4.

constraints, one obtains the grand canonical distribution

meaning of the chemical potential of species c.

6

the chemical potentials μ<sup>c</sup> to the particle densities nc,

<sup>ρ</sup>eq <sup>¼</sup> <sup>e</sup>

(normalization) and

Non-Equilibrium Particle Dynamics

maximum value for Sinf½ � ρ

The entropy Seqð Þ Ω; β; μ follows from Eq. (12). The dependence of extensive quantities on the volume Ω is trivial for homogeneous systems. After a thermodynamic potential is calculated, all thermodynamic variables are derived in a consistent manner. The method to construct statistical ensembles from the maximum of entropy at given conditions, which take into account the different contacts with the surrounding bath, is well accepted in equilibrium statistical mechanics and is applied successfully to different phenomena, including phase transitions.

Can we extend the definition of equilibrium entropy (12) also for ρð Þt that describes the evolution in nonequilibrium? Time evolution is given by an unitary transformation that leaves the trace invariant. Thus, the expression Trf g ρð Þt ln ρð Þt is constant for a solution ρð Þt of the von Neumann equation

$$\frac{d}{dt}[\operatorname{Tr}\{\rho(t)\ln\rho(t)\}]=\mathbf{0}.\tag{15}$$

The entropy for a system in nonequilibrium, however, may increase with time, according to the second law of thermodynamics. The equations of motion, including the Schrödinger equation and the Liouville-von Neumann equation, describe reversible motion and are not appropriate to describe irreversible processes. Therefore, the entropy concept (12) elaborated in equilibrium statistical physics together with the Liouville-von Neumann equation cannot be used as fundamental approach to nonequilibrium statistical physics.
