1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
63
nor concentration are uniform. In this case one realizes that the flow of heat gets a
contribution from the thermodynamic force due to the concentration gradient, with
proportionality constant L qm , and the diffusion of mass gets a contribution from the
thermodynamic force associated with the temperature gardient, with proportionality
constant L mq . It turns out that L qm = L mq . Similarly, the thermo-electric effect, or
Peltier-Seebeck-Thomson effect, couple heat flows with elctric currents.
Onsager relations introduce constraints on the conversion of heat to work, like
other thermodynamic relations, therefore their violation may in principle allow nondissipative thermodynamic currents. At the same time, it was argued, and universally
accepted, that the Onsager reciprocal relations do not hold in systems subjected to a
magnetic field, or a rotating reference frame, becasuse their dynamics are not TRI.
Casimir then argued the Onsager relations should be replaced by the following [38]:
L i j (B) = L ji (−B), i, j = 1, . . . , n
(1.225)
if B is the magnetic field in which the system of interest is immersed. While this
relation is conceptually satisfactory, it defeats the purpose of obtaining the transport
coeficient for a given system: two systems, one in the magnetic field B and the other
in the magnetic field –B, must be considered. Then, the L i j coefficient of one system
cannot be inferred from the L ji coefficient of that same system: a second experiment,
with a different system, must be performed.
However, non-dissipative currents have not been observed, so far, hence the
question whether magnetic fields break the Onsager symmetry has remained open.
Recently, this question has received an answer, in favour of the validity of Onsager
relations even in the presence of magnetic fields, or rotating frames, because in reality they do not break all possible time reversal symmetries [39]. This has direct
application on the correlation functions from which the transport coefficients can
be computed, via the Green-Kubo relations. In Ref. [39] it has been shown that for
systems whose particles interactions only depend on their relative positions, there
are at least 7 more time reversal operators that correspond to TRI. For instance, the
following operation:
i(x, y, z, p x , p y , p z ) = (x, −y, z, − p x , p y , − p z )
(1.226)
had been used in the case of shearing fluids. It preserves the form of the equations of
motion in presece of a magnetic field, if applied together with time inversion t → −t:
63
nor concentration are uniform. In this case one realizes that the flow of heat gets a
contribution from the thermodynamic force due to the concentration gradient, with
proportionality constant L qm , and the diffusion of mass gets a contribution from the
thermodynamic force associated with the temperature gardient, with proportionality
constant L mq . It turns out that L qm = L mq . Similarly, the thermo-electric effect, or
Peltier-Seebeck-Thomson effect, couple heat flows with elctric currents.
Onsager relations introduce constraints on the conversion of heat to work, like
other thermodynamic relations, therefore their violation may in principle allow nondissipative thermodynamic currents. At the same time, it was argued, and universally
accepted, that the Onsager reciprocal relations do not hold in systems subjected to a
magnetic field, or a rotating reference frame, becasuse their dynamics are not TRI.
Casimir then argued the Onsager relations should be replaced by the following [38]:
L i j (B) = L ji (−B), i, j = 1, . . . , n
(1.225)
if B is the magnetic field in which the system of interest is immersed. While this
relation is conceptually satisfactory, it defeats the purpose of obtaining the transport
coeficient for a given system: two systems, one in the magnetic field B and the other
in the magnetic field –B, must be considered. Then, the L i j coefficient of one system
cannot be inferred from the L ji coefficient of that same system: a second experiment,
with a different system, must be performed.
However, non-dissipative currents have not been observed, so far, hence the
question whether magnetic fields break the Onsager symmetry has remained open.
Recently, this question has received an answer, in favour of the validity of Onsager
relations even in the presence of magnetic fields, or rotating frames, because in reality they do not break all possible time reversal symmetries [39]. This has direct
application on the correlation functions from which the transport coefficients can
be computed, via the Green-Kubo relations. In Ref. [39] it has been shown that for
systems whose particles interactions only depend on their relative positions, there
are at least 7 more time reversal operators that correspond to TRI. For instance, the
following operation:
i(x, y, z, p x , p y , p z ) = (x, −y, z, − p x , p y , − p z )
(1.226)
had been used in the case of shearing fluids. It preserves the form of the equations of
motion in presece of a magnetic field, if applied together with time inversion t → −t: