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

i <sup>ℏ</sup> <sup>H</sup>t<sup>1</sup> <sup>F</sup> ; ρeq h i

Non-Equilibrium Particle Dynamics

we have Uð Þ <sup>t</sup>; <sup>t</sup><sup>1</sup> <sup>≃</sup> <sup>e</sup>�iHSð Þ <sup>t</sup>�t<sup>1</sup> <sup>=</sup>ℏ:

ρϵðÞ¼ <sup>t</sup> <sup>ρ</sup>relðÞ�<sup>t</sup> <sup>β</sup>e�iω<sup>t</sup>

þ ∑ n

> ∑ n

ð 1

0

h i A; B <sup>z</sup> ¼

ð Þ¼ AjB

eters Fn, finally we have

e�iω<sup>t</sup> .

36

¼ �βe

ð 1

dλ ∑ n

0

The last term in the curly bracket can be rewritten as

ρrel ¼ β

ð 0

dt<sup>1</sup> e�izt<sup>1</sup>

�∞

∂ ∂t1

conjugated parameters Fn (response parameters):

Bm; B\_ <sup>n</sup> � �

dλTr Ae�λβ<sup>H</sup> Beλβ<sup>H</sup> ρeq n o

> ð 0

�∞

The linear system of equations (162) has the form

∑ n

�iωt<sup>1</sup> ð 1

0

Because we restrict ourselves to the order Oð Þ h , for the time evolution operator

After linearization with respect to the external fields hj and the response param-

dλ � ∑ j

> Fn ¼ ∑ j

> > ∞ð

0

"

ð 1

0

FnB\_ <sup>n</sup>ð Þ <sup>i</sup>λβ<sup>ℏ</sup> <sup>þ</sup> <sup>t</sup><sup>1</sup> <sup>ρ</sup>eq � <sup>i</sup>ωFnBnð Þ <sup>i</sup>λβ<sup>ℏ</sup> <sup>þ</sup> <sup>t</sup><sup>1</sup> <sup>ρ</sup>eq � ��

(z ¼ ω þ iϵ). Here, we used that hjð Þt and Fnð Þt , Eq. (154), are proportional to

We multiply this equation by Bm, take the trace and use the self-consistency relation (151). We obtain a set of linear equations for the thermodynamically

with the Kubo scalar product (the particle number commutes with the observables)

¼ ð 1

0

<sup>z</sup> � iωh i Bm; B<sup>n</sup> <sup>z</sup> n o

and its Laplace transform, the thermodynamic correlation function

<sup>d</sup><sup>t</sup> <sup>e</sup>�iztð Þ¼ <sup>A</sup>jBð Þ<sup>t</sup>

PmnFn ¼ ∑

to determine the response parameters Fn, the number of equations coincides with the number of variables to be determined. The coefficients of this linear system of equations are given by equilibrium correlation functions. We emphasize

j

dλ ∑ j

hjA\_ <sup>j</sup>ð Þ <sup>i</sup>λβ<sup>ℏ</sup> <sup>ρ</sup>eq: (159)

<sup>F</sup>\_ <sup>n</sup>ð Þ <sup>t</sup><sup>1</sup> <sup>B</sup>nð Þ <sup>i</sup>λβ<sup>ℏ</sup> <sup>ρ</sup>eq: (160)

hj <sup>A</sup>\_ <sup>j</sup>ð Þ <sup>i</sup>λβ<sup>ℏ</sup> <sup>þ</sup> <sup>t</sup><sup>1</sup> <sup>ρ</sup>eq

Bm; A\_ <sup>j</sup> � �

dλTr AB ið Þ λβℏ ρeq n o

z

<sup>d</sup><sup>t</sup> <sup>e</sup><sup>i</sup>ztð Þ <sup>A</sup>ð Þj <sup>t</sup> <sup>B</sup> : (164)

Dmjhj (165)

hj, (162)

, (163)

(161)

that in the classical limit the relations become more simple because the variables commute, and we have not additional integrals expanding the exponential.

We can solve this linear system of equations (162) using Cramers rule. The response parameters Fn are found to be proportional to the external fields hj with coefficients that are ratios of two determinants. The matrix elements are given by equilibrium correlation functions. This way, the self-consistency conditions are solved, and the Lagrange multipliers can be eliminated. The nonequilibrium problem is formally solved. The second problem, the evaluation of equilibrium correlation functions, can be solved by different methods such as numerical simulations, quantum statistical perturbation theories such as thermodynamic Green functions and Feynman diagrams, path integral methods, etc. Using partial integration, we show the relation

$$-\text{iz}\langle \mathbf{A}; \mathbf{B} \rangle\_x = (\mathbf{A} \mid \mathbf{B}) + \langle \dot{\mathbf{A}}; \mathbf{B} \rangle\_x = (\mathbf{A} \mid \mathbf{B}) - \langle \mathbf{A}; \dot{\mathbf{B}} \rangle\_x. \tag{166}$$

Then, the generalized linear response equations (162) can be rewritten in the short form (165) with the matrix elements

$$P\_{mn} = \left(\mathbf{B}\_m|\dot{\mathbf{B}}\_n\right) + \left<\dot{\mathbf{B}}\_m; \dot{\mathbf{B}}\_n\right>\_{oo+ic} - \mathrm{i}o\left(\mathbf{B}\_m|\mathbf{B}\_n\right) - \mathrm{i}o\left<\dot{\mathbf{B}}\_m; \mathbf{B}\_n\right>\_{oo+ic} \tag{167}$$

$$D\_{mj} = \left(\mathbf{B}\_m|\dot{\mathbf{A}}\_j\right) + \left<\dot{\mathbf{B}}\_m; \dot{\mathbf{A}}\_j\right>\_{\boldsymbol{\alpha} + i\epsilon}.\tag{168}$$

that can be interpreted as generalized transition rates (collision integral, left-hand side) and the influence of external forces (drift term, right-hand side of Eq. (165)).

Having the response parameters Fn to our disposal, we can evaluate averages of the relevant observables, see Eq. (151),

$$\left\langle \mathbf{B}\_{n} \right\rangle^{t} = \left\langle \mathbf{B}\_{n} \right\rangle\_{\text{rel}}^{t} = -\beta \sum\_{m} F\_{m} \mathbf{e}^{\text{lat}} N\_{mn} \qquad N\_{mn} = \left( \mathbf{B}\_{m} \middle| \mathbf{B}\_{n} \right). \tag{169}$$

Eliminating Fm, these average fluctuations Bh i<sup>n</sup> <sup>t</sup> are proportional to the fields hje�iω<sup>t</sup> .
