1.3.3 Principle of Minimum Entropy Production
Main problem of non-equilibrium thermodynamics is to extend method of thermodynamics covering non-linear and unstable phenomenon by start from equilibrium
state. Principle of local equilibrium is postulated for this purpose. Considering
volume element of small enough but sufficient enormous size for macroscopic nature
in non-equilibrium state including irreversible process, there exist state valuables of
state function thermodynamics. The state function is as same as that in equilibrium
thermodynamics. Principle of local equilibrium implies that effect of collision is
dominant to keep thermodynamically. It becomes possible to extend to
non-equilibrium state if local entropy is described by same independent valuable.
Substances and energies exchange through its surfaces among of each subsystem.
And the surface increases at second power while the volume increases at third power
as the size of system increases. Therefore, the large system enough to establish
condition of statistical independency can be regarded as closed system. And those
subsystems are independent from surrounding subsystems. Ergodic hypothesis is
available and statistical method can be used in those subsystems.
To introduce ‘Principle of minimum entropy production’, ‘Law of conservation
of entropy’ is first introduced from ‘Principle of local equilibrium’. In General
expression of conservation law, specific entropy e is introduced into a. And it
becomes equation,
d
dt ¼
∂
∂t
þ v∇. By coordinates transformation on flow and then
use of equation of continuity
∂ρ a
∂t
þ ∇J s À φ s ¼ 0, equation entropy balance,
ρ
dS
dt ¼ ΦT À ∇j s in non-equilibrium state is obtained. In general, inner generation
rate of entropy production, Φ is expressed as Φ ¼ ρT
dS
dt ¼
P
J i X i by definition of
flux J and force X.
In the presence of n fluxes, some of them are zero in stationary state. When
phenomenon equation J i ¼
P
j
L ij X j is assumed, generation rate of entropy production is expressed as Φ ¼
P
i
P
j L ij X i X j and derivation of the equation is described in
the next section. When forces X 1 , X 2 , . . ., X k are fixed, condition of Φ having
maxima or minima is
∂Φ
∂X i
¼ 0 i ¼ k þ 1; . . . ; n
ð
Þ . Substituting Φ for Φ ¼
P
i
P
j L ij
X i X j gives
X n
i¼1
L ij þ L ji
À
Á ¼ 0
i ¼ k þ 1; . . . ; n
ð
Þ . And Ji ¼ 0 is obtained by using
Onsager reciprocal relations, L ij ¼ L ji (i 6 ¼ j). The result indicates that forces
conjugating the fixed forces remain and other forces become zero. The fixed forces
maintain stationary state. In zero-order stationary state, there is no fixed force and all
fluxes disappear, and this is equilibrium state. When one of unfixed forces is
perturbed, positive flux caused by the force generates negative flux of resistance,
and this is an extended Le Chaterier.’s principle. Principle of minimum production
of entropy is an important achievement obtained in non-equilibrium thermodynamics. The principle indicates that system selects a path of minimum resistance.
16
1 Physics in the Origin of Life
Main problem of non-equilibrium thermodynamics is to extend method of thermodynamics covering non-linear and unstable phenomenon by start from equilibrium
state. Principle of local equilibrium is postulated for this purpose. Considering
volume element of small enough but sufficient enormous size for macroscopic nature
in non-equilibrium state including irreversible process, there exist state valuables of
state function thermodynamics. The state function is as same as that in equilibrium
thermodynamics. Principle of local equilibrium implies that effect of collision is
dominant to keep thermodynamically. It becomes possible to extend to
non-equilibrium state if local entropy is described by same independent valuable.
Substances and energies exchange through its surfaces among of each subsystem.
And the surface increases at second power while the volume increases at third power
as the size of system increases. Therefore, the large system enough to establish
condition of statistical independency can be regarded as closed system. And those
subsystems are independent from surrounding subsystems. Ergodic hypothesis is
available and statistical method can be used in those subsystems.
To introduce ‘Principle of minimum entropy production’, ‘Law of conservation
of entropy’ is first introduced from ‘Principle of local equilibrium’. In General
expression of conservation law, specific entropy e is introduced into a. And it
becomes equation,
d
dt ¼
∂
∂t
þ v∇. By coordinates transformation on flow and then
use of equation of continuity
∂ρ a
∂t
þ ∇J s À φ s ¼ 0, equation entropy balance,
ρ
dS
dt ¼ ΦT À ∇j s in non-equilibrium state is obtained. In general, inner generation
rate of entropy production, Φ is expressed as Φ ¼ ρT
dS
dt ¼
P
J i X i by definition of
flux J and force X.
In the presence of n fluxes, some of them are zero in stationary state. When
phenomenon equation J i ¼
P
j
L ij X j is assumed, generation rate of entropy production is expressed as Φ ¼
P
i
P
j L ij X i X j and derivation of the equation is described in
the next section. When forces X 1 , X 2 , . . ., X k are fixed, condition of Φ having
maxima or minima is
∂Φ
∂X i
¼ 0 i ¼ k þ 1; . . . ; n
ð
Þ . Substituting Φ for Φ ¼
P
i
P
j L ij
X i X j gives
X n
i¼1
L ij þ L ji
À
Á ¼ 0
i ¼ k þ 1; . . . ; n
ð
Þ . And Ji ¼ 0 is obtained by using
Onsager reciprocal relations, L ij ¼ L ji (i 6 ¼ j). The result indicates that forces
conjugating the fixed forces remain and other forces become zero. The fixed forces
maintain stationary state. In zero-order stationary state, there is no fixed force and all
fluxes disappear, and this is equilibrium state. When one of unfixed forces is
perturbed, positive flux caused by the force generates negative flux of resistance,
and this is an extended Le Chaterier.’s principle. Principle of minimum production
of entropy is an important achievement obtained in non-equilibrium thermodynamics. The principle indicates that system selects a path of minimum resistance.
16
1 Physics in the Origin of Life