N 1 /NFraction
7.8 Onsager Reciprocity Relations For a Modified Dog-Flea ModelModel
In Chapter 2 we briefly discussed about the Onsager reciprocal relations, which relates the coupling between two different types of fluxes. We also briefly mentioned in Chapter 6 that these relations fall out naturally from the maximum caliber principle [7, 8]. In this section we explore the simplest dynamical model in which there is a coupling of two different types of flows that can be explored both near equilibrium and far from equilibrium. Our model is a variant of the earlier used dog-flea model, which we will use to treat the problem of coupled dynamical flows by the application of the principle of maximum caliber.
7.8.1 The hot-dog model and the principle of maximum caliber
As before, consider dogs 1 and 2 with N1 and N2 fleas, respectively, at time t. The fleas here are of two types: the ground-statefleas (GS) or theexcited-statefleas (ES). The ES fleas are higher in energy than GS fleas by an amount. A flea can be in one of the four possible states: 1GS,1ES,2GS, or 2ES, depending on whether it sits on dog-1 or dog-2 and if it is in GS or ES. The fleas on dog-1 are at temperatureT1, and those on dog-2 at temperatureT2 (see Figure 7.10). As the present model differs from the simpler Dog-Flea model in our inclusion of thermal excitation, we call this the Hot Dog (HD) Model.
m1
N1(t) N2(t)
n
2m
2m
1j1 j2
n
1 i2i1
T
1T
2Figure 7.10: Schematic of the simple 2-state “hot-dog” model. On each dog the fleas can be in two states, the excited state (ES) and the ground state (GS). The total number of fleas in the ES of dog-1 and dog-2 (red fleas) arem1 and m2, respectively, while the total number of fleas in the GS of dog-1 and dog-2 (black fleas), respectively, are n1 andn2 respectively. The red fleas are greater in energy than the black fleas by an amount . The temperatures of the fleas on dog-1 and dog-2 areT1andT2, respectively.
The occupancy number of the fleas in each state are denoted by the following convention:
• m1 ES fleas are on Dog 1.
• n1 GS fleas are on Dog 1.
• m2 ES fleas are on Dog 2.
• n2 GS fleas are on Dog 2.
The total number of fleas on dog-1 and dog-2 are, respectively, given by
N1=n1+m1 (7.53)
and
N2=n2+m2. (7.54)
Further, the GS and ES fleas are in thermal equilibrium on each dog, i.e, for dog-1 m1
n1 = exp(−/kT1) (7.55)
and for dog-2
m2
n2
= exp(−/kT2). (7.56)
We wish to obtain equations for the transport of fleas between the two dogs. A flea on a given dog can jump onto another dog within a time interval ∆t according to a few simple rules:
• In the time-interval ∆ta flea can either stay put on its current dog, or jump to the other dog.
• In the case of a jump, an ES (GS) flea remains an ES (GS) flea within the time-interval ∆t.
• A flea can change its thermal state only after it has arrived on a dog, where it then equilibrates rapidly (compared to the jump rate) between the ES and the GS.
• Theintrinsic jump rateandintrinsic stay rate ispand 1−p, respectively, for the fleas.
We now adopt Jaynes’ principle of maximum caliber for computing the dynamics of the model.
To evaluate the space of trajectories for the model we introduce the following variables:
• i1fleas jump from the GS of dog-1 to the GS of dog-2.
• j1fleas jump from the ES of dog-1 to the ES of dog-2.
• i2fleas jump from the GS of dog-2 to the GS of dog-1.
• j2fleas jump from the ES of dog-2 to the ES of dog-1.
The rest of the fleas stay put on their respective dogs. Thus, a given combination of (i1, j1, i2, j2) defines a trajectory Γ of duration ∆tfor the model. As in the previous sections, we maximize the entropy over the possible trajectories. We subject subject the entropy to two constraints:
1. Average particle flux constraint: The particle flux can be resulted by the flux of both ES and GS particles. The average particle flux isP
ΓpΓ(i1+j1−i2−j2).
2. Average heat-flux constraint: Since the heat flux is possible only by the motion of ES fleas, the average heat flux is P
ΓpΓ(j1−j2).
As explained in the previous sections, we now evaluate the intrinsic jump-rate prior:
• The prior for the jump ofii GS particles from dog-1 to dog-2 ispi1(1−p)(n1−i1).
• The prior for the jump ofj1 ES particles from dog-1 to dog-2 ispj1(1−p)(m1−j1).
• The prior for the jump ofi2 GS particles from dog-1 to dog-22 ispi2(1−p)(n2−i2).
• The prior for the jump ofj2 ES particles from dog-1 to dog-2 ispj2(1−p)(m2−j2).
The total probability prior for the trajectory will be a product of all these expressions. The caliber for the process can now be written as
C =X
Γ
pΓln
pΓ
pi1(1−p)(N1−i1)pj1(1−p)(m1−j1)pi2(1−p)(N2−i2)pj2(1−p)(m2−j2)
−λe
X
Γ
pΓ(j1−j2)−λd
X
Γ
pΓ(i1+j1−i2−j2),
where Γ corresponds to the combination (i1, i2, j1, j2). The multipliers λd and λe correspond to the “average particle flux” and the “average heat flux,” respectively. The partition function can be evaluated as
Z = (1−p+pexp(λd+λe))m1(1−p+pexp(λd))n1(1−p+pexp(−λd−λe))m2×...
× (1−p+pexp(−λd))n2 (7.57)
The average flux of fleas is given by
hJdi= ∂lnZ
∂λd
. (7.58)
while the average flux of heat is given by
hJei=∂lnZ
∂λe (7.59)
Thus,
hJdi = m1pexp(λd+λe)
1−p+pexp(λd+λe)+ n1pexp(λd)
1−p+pexp(λd)− m2pexp(−λd−λe) 1−p+pexp(−λd−λe)...
− n2pexp(−λd)
1−p+pexp(−λd) (7.60)
and
hJei= m1pexp(λd+λe)
1−p+pexp(λd+λe)− m2pexp(−λd−λe)
1−p+pexp(−λd−λe). (7.61) Within the limitations of our model, these expressions for flux are completely general and apply arbitrarily far from equilibrium. As explain in Chapter 6, we interpret the Lagrange multipliersλd
andλeas the driving forces for the particle and the heat diffusion, respectively.
7.8.2 Proving the Onsager Reciprocal Relations
The reasons for choosing this particular model here are:
1. It applies not just near equilibrium, but rather, we can get exact expressions for its behavior over the full nonequilibrium range, including far from equilibrium.
2. We believe that this is the very simplest microscopic model for coupled flow dynamics, which is essential if we want to study the reciprocal relations.
We first consider the behavior of the model near equilibrium, which has been the main focus of much of the past work with the reciprocal relations [25]. The average flux of particles, hJd(λd, λe)i and the average flux of energy, hJe(λd, λe)i, can each be expressed as a function of the two Lagrange multipliers,λd and λe, that represent forces acting on the system (see below). We expand around λd= 0 andλe= 0 to obtain:
hJdi=hJd(0,0)i+ ∂hJdi
∂λd
0,0
λd+ ∂hJdi
∂λe
0,0
λe (7.62)
hJei=hJe(0,0)i+ ∂hJdi
∂λd
0,0
λd+ ∂hJei
∂λe
0,0
λe. (7.63)
Written in the Onsager form, we have
hJdi=hJd(0,0)i+L11λd+L12λe (7.64)
and
hJei=hJe(0,0)i+L21λd+L22λe. (7.65)
The virtues of the maximum caliber principle here are: (1) that it makes deriving the reciprocal relations very simple and (2) it shows the direct resemblance of the reciprocal relations of dynamics to Maxwell’s relations of equilibrium thermodynamics [25]. From equations 7.58 and 7.59, we have
L12= ∂hJdi
∂λe 0,0
= ∂2lnZ
∂λe∂λd 0,0
(7.66)
and
L21= ∂hJei
∂λd
0,0
= ∂2lnZ
∂λd∂λe
0,0
(7.67)
Thus, due to the equality of mixed derivatives, it can be seen from Eqs. 7.66 and 7.67 thatL12=L21. Within the present model, the Onsager reciprocal relation, L12=L21, also holds arbitrarily far from equilibrium. This is readily obtained from the following Maxwell-like relationship:
L12= ∂hJdi
∂λe
= ∂2lnZ
∂λe∂λd
(7.68) and
L21= ∂hJei
∂λd = ∂2lnZ
∂λd∂λe (7.69)
7.8.3 Obtaining the Driving Forces for the Model
In the previous section we provided a simple derivation of the Onsager reciprocal relations. Here, we will evaluate the corresponding kinetic coefficients in terms of the parameters of the underlying microscopic model. Using Eq. 7.60, we have
L11= ∂hJdi
∂λd
0,0
= m1p−m1p2+n1p−n1p2+m2p−m2p2+n2p−n2p2
= p(1−p)(m1+n1+m2+n2) = 2p(1−p)(N1+N2),
= 2p(1−p)N, (7.70)
where we have used N1+N −2 to be equal toN, the total number of fleas on dog-1 and dog-2 combined. Now, to evaluateL12we use Eq 7.61 to obtain
L12= ∂hJdi
∂λe
0,0
=p(1−p)(m1+m2). (7.71)
We saw earlier thatL12 equalsL21 and hence do not need to evaluate it separately. To obtainL22, we use equation 7.61:
L22= ∂hJei
∂λe
0,0
=2p(1−p)(m1+m2). (7.72)
Thus we have obtained the values of the kinetic coefficients for our simple hot-dog model.