Theory of the linear answer

In Physical statistics out of balance, the theory of the linear answer make it possible to define susceptibilities and the coefficients of transport of a system in the vicinity of thermal balance independently of the details of the model. The theory of the linear answer was developed in the years 1950 by Greene, H.B. Callen, Ryogo Kubo.

General formalism

The Hamiltonian of the system

In the theory of the linear answer, one supposes that the system considered is described by a certain Hamiltonian of balance $H\_0$, disturbed by a perturbatif Hamiltonian depend on time $H\_1\; (T)$, that one can clarify in the form:
$H\_1\; (T)\; =\; \backslash \; sum\_i\; \backslash \; lambda\_i\; (T)\; A\_i$
or the $\backslash \; lambda\_i\; (T)$ are the factors perturbatifs and the square operators $A\_i$ are observable system, so that the total Hamiltonien of the system is: $H=H\_0+H\_1\; (T)\; =H\_0+\; \backslash \; sum\_l\; \backslash \; lambda\_i\; (T)\; A\_i$ This reveals that the natural formalism of a system in linear answer is the Représentation of interaction.

Clarification of the matrix density

The Matrice density is of course affected by the disturbance of the Hamiltonian. We point out that for the matrix density, the equation of Schrödinger is written: Caution! This is not a equation of Heisenberg and the Matrice density $\backslash \; rho\; (T)$ is not an operator! (Even the sign of the switch to be convinced some!) If we baptize $\backslash \; rho\_0$ the matrix density of the nondisturbed system (i.e. system with thermal balance), $\backslash \; rho\; (T)$ the matrix density of the disturbed system (i.e. system except balance), $\backslash \; delta\; \backslash \; rho\; (T)$ the variation of the matrix density of the disturbed system calculated with the first order of disturbance, the equation of the matrix density is reduced to:
$I\; \backslash \; hbar\; \backslash \; partial\_t\; \backslash \; delta\; \backslash \; rho\; (T)\; =\; +$
from where the solution:
$\backslash \; delta\; \backslash \; rho\; (T)\; =\; \{1\; \backslash \; over\; \{I\; \backslash \; hbar\}\}\; \backslash \; int\_\; \{-\; \backslash \; infin\}\; ^t\; e^\; \{-\; I\; (T\; \backslash \; tau)\; H\_0/\backslash \; hbar\}\; e^\; \{I\; (T\; \backslash \; tau)\; H\_0/\backslash \; hbar\}\; D\; \backslash \; tau$
what gives access $\backslash \; rho\; (T)$

calculation of observable and the functions of answer

While calculating with the first order of the Théorie of the disturbances one obtains the Matrice density $\backslash \; rho\; (T)$ of the system. This matrix can be used to extract average thermics and quantum from the observable ones:

$\backslash \; langle\; A\_k\; \backslash \; rangle\; (T)\; =Tr\; (\backslash \; rho\; (T)\; A\_k)$

In last spring, by introducing the function of delayed answer $\backslash \; chi\_\; \{kl\}\; (T)$ the observable ones of the system is given by:

Where one identifies the function of answer $\backslash \; chi\_\; \{kl\}\; (T)$ by: where $\backslash \; theta$ is the Fonction of Heaviside (which translates here the Law of causality), $A\_k\; (T)\; =\; \backslash \; exp\; (I\; H\_0\; T\; \backslash \; hbar)\; A\_k\; \backslash \; exp\; (-\; I\; H\_0\; T\; \backslash \; hbar)$ is the operators in the representation of Heisenberg, and the average is taken with the matrix density of balance $\backslash \; rho\_0=e^\; \{-\; \backslash \; beta\; H\_0\}\; /Tr\; (e^\; \{-\; \backslash \; beta\; H\_0\})$. The fact that function of answer depends only on the difference in time between the excitation and the measurement of the answer is a consequence of the catch of average on a state of balance, which is invariant by translation in time.

The definition of the function of answer is due to Ryogo Kubo (1956).

As the function of answer $\backslash \; chi\_\; \{kl\}\; (T)$ is cancelled for (because of the law of causality), one can define his transformed of imaginary Laplace (still named Fourrier unilateral), which is equal in this precise case to its transform of simple Furrier:

$\backslash \; chi\_\; \{kl\}\; (Z)\; =\; \backslash \; int\_0^\; \backslash \; infty\; \backslash \; chi\_\; \{kl\}\; (T)\; e^\; \{I\; Z\; T\}\; dt=\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \backslash \; infty\; \backslash \; chi\_\; \{kl\}\; (T)\; e^\; \{I\; Z\; T\}\; dt$

who is thus a holomorphic Fonction for $\backslash \; mathrm\; \{Im\}\; (Z)\; >0$ according to the properties of the transformation of Laplace.

First application: electrical resistance

The interest of the theory of the linear answer comes from what no assumption is necessary on the Hamiltonian $H\_0$ to define the function of answer. That allows for example to define conductivity while considering:

$H\_0\; -\; \backslash \; vec\; \{J\}\; \_0\; \backslash \; cdot\; \backslash \; vec\; \{has\}\; (T)$

where $\backslash \; vec\; \{J\}$ is the electric current, and $\backslash \; vec\; \{has\}$ is the potential vector. The theory of the linear answer gives a relation then:

$\backslash \; vec\; \{J\}\; \_0\; (\backslash \; Omega)\; =\; \backslash \; chi\_\; \{JJ\}\; (\backslash \; Omega)\; \backslash \; vec\; \{has\}\; (\backslash \; Omega)$

By taking account of the Maxwell's equations, this equation makes it possible to show that conductivity is:

$\backslash \; sigma\; (\backslash \; Omega)\; =\; \backslash \; frac\; \{\backslash \; chi\_\; \{JJ\}\; (\backslash \; Omega)\}\{I\; \backslash \; Omega\}\; +\; \backslash \; frac\; \{N\; e^2\}\; \{m\; I\; \backslash \; Omega\}$

The second term is a diamagnetic contribution which comes from what the current is $\backslash \; vec\; \{J\}\; \_0-ne^2\; \backslash \; vec\; \{has\}\; /m$ in the presence of a potential vector.

The calculation of conductivity is thus tiny room to the calculation of the function of answer $\backslash \; chi\_\; \{JJ\}$. This calculation can be carried out either by numerical methods like the quantum Monte Carlo method, or method of Lanczos or by analytical methods like the summation of diagrams of Feynman.

Another application: magnetic relieving

In the same way, one can define with the theory of the linear answer of other physical sizes like the permittivity or magnetic susceptibility. Magnetic susceptibility is in particular useful in the study of electronic paramagnetic resonance.

Within the framework of the theory of the linear answer, it is also possible to study the processes of relieving by calculating the answer to a disturbance of the form:

$\backslash \; lambda\; (T)\; =\; \backslash \; lambda\_0\; \backslash \; exp\; (\backslash \; epsilon\; T)\; \backslash \; theta\; (-\; T)$

and by taking the limit $\backslash \; epsilon\; \backslash \; to\; 0$.

Thus, the theory of the linear answer makes it possible to define the relaxation time $1/T\_1$ resulting from the hyperfine coupling between the nuclear spins and the electronic spins without making of assumption a priori on the model which describes the electronic spins.

Lastly, the theory of the linear answer allows thanks to the Théorème of fluctuation-dissipation of to define the functions of answer in term of the symmetrical functions of correlation:

$S\_\; \{lk\}\; (T-you)\; =\; \backslash \; frac\; 1\; 2\; \backslash \; langle\; A\_l\; (T)\; A\_k\; (you)\; +\; A\_k\; (you)\; A\_l\; (T)\; \backslash \; rangle$

Nonmechanical disturbances

In what precedes, it was admitted that the function of answer could be obtained while calculating evolution of a system whose Hamiltonian depends explicitly on time by the theory on disturbances. In this case, one speaks about mechanical disturbances.

However, if one wants to be able to define quantities as thermal conductivity or the constant of diffusion, this framework is too restrictive. Indeed, a heat gradient cannot be seen like a force acting on the particles of a system. One speaks then about nonmechanical disturbances. In the case of thermal transport, a generalization of the formula of Kubo was proposed by J. Mr. Luttinger in 1964. This generalization is based on an assumption of local balance.

Linear answer and relations of reciprocity of Onsager

The theory of the linear answer makes it possible to give a microscopic justification of the relations of reciprocity of Onsager. One obtains in fact a more general equality:

$\backslash \; mathrm\; \{Im\}\; \backslash \; chi\_\; \{lk\}\; (\backslash \; Omega)\; =\; \backslash \; mathrm\; \{Im\}\; \backslash \; chi\_\; \{kl\}\; (\backslash \; Omega)$

if the operators $A\_l$ and $A\_k$ are all the two invariants by inversion of the time and where the system is not placed in a magnetic field. When the system is placed in a field it is necessary to change the sign of the magnetic field in the member of right-hand side of the equality. If the operators $A\_l$ or $A\_k$ change sign under inversion of direction of time (for example if they are two currents), it is necessary to apply the same number of changes of sign to the member of right-hand side as operators not invariants by inversion of the direction of time (in the case of two currents, one must apply two changes of sign, therefore the final sign does not change in the member of right-hand side).

Relations of Kramers-Kronig

The fact that the function of correlation is cancelled for the time intervals negative is a consequence of causality. Indeed, that means that at time $t$, the answer system depends only on the values of the disturbance at times

$\backslash \; chi\; (Z)\; =\; \backslash \; frac\; \{1\}\; \{2i\; \backslash \; pi\}\; \backslash \; int\; D\; \backslash \; omega\text{'}\; \backslash \; frac\; \{\backslash \; chi\; (\backslash \; omega\text{'})\}\{Z\; \backslash \; omega\text{'}\}$ By making $z\; \backslash \; to\; \backslash \; omega+i\; 0$ and by using the identities on the distributions, one obtains the relations of Kramers-Kronig:

$\backslash \; mathrm\; \{Re\}\; \backslash \; chi\; (\backslash \; Omega)\; =\; \backslash \; int\; \backslash \; frac\; \{D\; \backslash \; omega\text{'}\}\; \{\backslash \; pi\}\; \backslash \; frac\; \{\backslash \; mathrm\; \{Im\}\; \backslash \; chi\; (\backslash \; omega\text{'})\}\{\backslash \; Omega\; \backslash \; omega\text{'}\}$ $\backslash \; mathrm\; \{Im\}\; \backslash \; chi\; (\backslash \; Omega)\; =\; \backslash \; int\; \backslash \; frac\; \{D\; \backslash \; omega\text{'}\}\; \{\backslash \; pi\}\; \backslash \; frac\; \{\backslash \; mathrm\; \{Re\}\; \backslash \; chi\; (\backslash \; omega\text{'})\}\{\backslash \; Omega\; \backslash \; omega\text{'}\}$

Rules of nap

The rules of nap are identities satisfied by the functions with answer of the form: $\backslash \; int\; D\; \backslash \; Omega\; \backslash \; omega^n\; \backslash \; chi\_\; \{lk\}\; (\backslash \; Omega)\; =\; C\_n$

Where $C\_n$ is the median value of a certain operator in the state of balance. These rules of nap are obtained by integrating by part the formula of transformation of Laplace. Integration by part reveals derivative of the operator $A\_\; \{L\}$ who can to be represented using the equation of the movement of Heisenberg. One obtains as follows: $C\_n\; =\; \backslash \; langle\; \backslash \; rangle$.

Formalism of Mori-Zwanzig

The formalism of Mori-Zwanzig uses operators of projection to obtain expressions functions of opposite answer. For example, the formalism of Mori-Zwanzig makes it possible to obtain an expression for the resistivity rather than conductivity.

References

• R. Kubo J. Phys. Plowshare Jpn 12 , 570 (1957).
• L.P. Kadanoff and PC Martin Anarchist. Phys. (NY) 24 , 419 (1963).
• J. Mr. Luttinger Phys. rev. 135 , A1505 (1964).
• R. Zwanzig Phys. rev. 123 , 983 (1961); J. Chem. Phys. 33 , 1338 (1960).
• H. Mori Phys. rev. 112 , 1829 (1958); Prog. Theor. Phys. (Kyoto) 33 , 423 (1965).
• L.D. Landau and E. Mr. Lifshitz Course of Theoretical physics T. 5, Physical Statistics (Mir/Ellipses)
• R. Kubo, Toda, Hashistume Statistical Physics II (Springer-Verlag)
• D. Forster Hydrodynamic fluctuations, broken symmetries and correlation functions (Benjamin Cummings)
• http://cel.ccsd.cnrs.fr/cel-00092930 (Course of DEA on the phenomena except balance by NR. Pottier)
• http://cel.ccsd.cnrs.fr/cel-00092959 (Course of DEA on the phenomena except balance by P. - has. Martin)

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