J. Phys. Chem. 1996, 100, 19035-19042
19035
Fokker-Planck Equation and Langevin Equation for One Brownian Particle in a Nonequilibrium Bath Joan-Emma Shea and Irwin Oppenheim*,† Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ReceiVed: June 3, 1996X
The Brownian motion of a large spherical particle of mass M immersed in a nonequilibrium bath of N light spherical particles of mass m is studied. A Fokker-Planck equation and a generalized Langevin equation for an arbitrary function of the position and momentum of the Brownian particle are derived from first principles of statistical mechanics using time-dependent projection operators. These projection operators reflect the nonequilibrium nature of the bath, which is described by the exact nonequilibrium distribution function of Oppenheim and Levine [Oppenheim, I.; Levine, R. D. Physica A 1979, 99, 383]. The Fokker-Planck equation is obtained by eliminating the fast bath variables of the system [Van Kampen, N. G.; Oppenheim, I. Physica A 1986, 138, 231], while the Langevin equation is obtained using a projection operator which averages over these variables [Mazur, P.; Oppenheim, I. Physica 1970, 50, 241]. The two methods yield equivalent results, valid to second order in the small parameters ) (m/M)1/2 and λ, where λ is a measure of the magnitude of the macroscopic gradients of the system.
1. Introduction Brownian motion is perhaps the best known example of a stochastic process in physics. Systems consisting of one Brownian particle in an equilibrium bath have been extensively studied1-4 and are well understood. Treatments of systems in which the bath is not in equilibrium have however been less successful, and studies have generally been limited to fairly simple models in which gradients only in either temperature5,6 or velocity7-11 are considered. In this paper, we propose to deal with the problem of a Brownian particle in a nonequilibrium bath in a very general manner, by describing the bath particles by the exact nonequilibrium distribution function derived by Oppenheim and Levine.1 This model has the advantage of including the hydrodynamic variables of the bath and of enabling us to account for gradients in pressure, velocity, and temperature. Our treatment is based on the use of time-dependent projection operators and on expansions in the small parameters ) (m/M)1/2 and λ. We use the expansion in to separate the time scales of the Brownian and bath particles. The bath particles relax fairly quickly to a state of local equilibrium, and we will thus express all the correlation functions in terms of local equilibrium averages. We shall further use the expansion in λ to rewrite the local equilibrium averages as homogeneous local equilibrium averages.1,15,16 Our paper is organized as follows. In section 2, we shall derive the Fokker-Planck equation for the Brownian particle using the van Kampen method of elimination of fast variables.2 We shall generalize the van Kampen projection operator to a time-dependent projection operator that takes into account the nonequilibrium nature of the bath. The new dissipative terms appearing in the Fokker-Planck equation will be discussed in section 3, and our results will be compared to those of other researchers in section 4. In section 5, we shall use the Hermitian conjugate of the projection operator of section 2 to derive the generalized Langevin equation for an arbitrary function of the position and momentum of the Brownian particle. The deriva* To whom all correspondence should be addressed. † This paper is dedicated to John Ross with great respect and deep admiration for our many years of friendship and scientific interactions. X Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01605-X CCC: $12.00
tion of the Langevin equation follows the treatment of Mazur and Oppenheim.14 We shall then compute an average Langevin equation from the Fokker-Planck equation and show that our two methods of treating Brownian motion are in fact equivalent. 2. Fokker-Planck Equation 2.1. Hamiltonian and Liouville Operator. We consider a system consisting of one heavy Brownian particle of mass M immersed in a nonequilibrium bath of N light particles of mass m. The coordinates and momenta of the Brownian particle will be denoted by XB ) (R,PB) and those of the bath by X ) (rN,pN). The densities of the Brownian particle and the bath fluid are similar, and all the interactions are short-ranged. The Hamiltonian H(X,XB) for the system can be written as
H(X,XB) ) HB(XB) + H0(X,R)
(2.1)
where
PB‚PB 2M
(2.2)
pN‚pN + U(rN) + V(rN,R) 2m
(2.3)
HB(XB) ) and
H0(X) )
The Hamiltonian H0(X) is the Hamiltonian for the fluid in the presence of the Brownian particle held fixed at the position R. The bath-bath and bath-Brownian interactions are described by the potentials U(rN) and V(R,rN), respectively. They are both sums of two-body terms and are given by N
U(rN) ) ∑ ∑ u(|ri - rj|)
(2.4)
i)1 j>i
and N
V(R,rN) ) ∑ w(|R - ri|) i)1
© 1996 American Chemical Society
(2.5)
19036 J. Phys. Chem., Vol. 100, No. 49, 1996
Shea and Oppenheim
It is convenient to define reduced momenta for both the Brownian and the bath particle
PB ) PB† + MvB(R)
(2.6)
P*B† ) PB†
(2.7)
pj† ) pj - mv(rj)
(2.8)
and
where
C(r) )
( ) 1 Aˆ (r)
and Aˆ (r) ) A(r) - 〈A(r)〉t. The brackets 〈...〉t denote a local equilibrium average over the distribution function σ(t). The variables A(r) form a special set consisting of the number density N(r), the momentum density P(r), and the energy density E(r) of the bath. The bath densities are given by the following expressions
where ) (m/M)1/2 is a small parameter that reflects the difference in masses of the Brownian and bath particles. The quantity vB(R) corresponds to the average velocity of the Brownian particle. The Liouvillian of the system is given by
N(r) ) ∑ δ(r - ri)
L ) L0 + LB
P(r) ) ∑ piδ(r - ri)
N
(2.18)
i)1
N
(2.9)
(2.19)
i)1
where N
L0 ) -
E(r) ) ∑ eiδ(r - ri)
N
p N ‚∇ + ∇Nr (U + V)‚∇PN m r
(2.10)
(2.20)
i)1
where
and
PB LB ) - ‚∇R + ∇RV‚∇PB M
ei )
(2.11)
pi‚pi
+
2m
1
N
∑
2 j)1,j*i
u(|ri - rj|) + w(|R - ri|)
(2.21)
In terms of the reduced notation, the Liouvillian can be expressed as
φ(r,t) is a vector whose components are the forces conjugate to the dynamical variables C(r).1,15,16
L ) L′ + LB†
φ1(r,t) ) 0
(2.22)
φN(r,t) ) β(r,t)[µ(r,t) - 1/2mv2(r,t)]
(2.23)
φP(r,t) ) β(r,t) v(r,t)
(2.24)
φE(r,t) ) -β(r,t)
(2.25)
(2.12)
where
L′ ) L0 - vB(R)‚∇R
(2.13)
and † P* B † † LB ) ‚∇ + ∇RV‚∇P* B m R
(2.14)
2.2. Distribution Functions. The Liouvillian L governs the dynamics of the distribution function F(t) of the total system
F˙(t) ) ∂F(t)/∂t ) LF(t)
(2.15)
We will use projection operator techniques to derive a FokkerPlanck equation for the reduced distribution function W(XB,t) of the Brownian particle. W(XB,t) is given by
(2.16) N
where the trace operation Tr involves an integration over the phase space of the bath and a summation over the number of bath particles. The local equilibrium distribution function for the bath in the presence of a fixed Brownian particle is given by1,15,16
(
) (
C(r)*φ(r,t) ) A(r)*φ(r,t) ) N(r)*(βµ)(r,t) + E†(r)*(-β)(r,t) (2.26) where
W(XB,t) ) Tr[F(t)]
1 A(r)*φ(r,t) 1 C(r) φ(r,t) e * e 3N N!h N!h3N ) σ(t) ) 1 C(r) φ(r,t) 1 A(r) φ(r,t) Tr e * Tr e * 3N N!h N!h3N
β(r,t) ) 1/kBT(r,t) and T(r,t), µ(r,t), and v(r,t) are the local temperature, chemical potential, and velocity, respectively. The * in C(r)*(r,t) denotes a scalar product, an integration over the spatial argument r, and a summation over the hydrodynamic variables. C(r)*φ(r,t) can be written in the reduced momentum notation as
E†(r) ) ∑ ei†δ(r - ri) and
ei†
)
(2.17)
(2.27)
i)1
)
pi†‚pi† 2m
+
1
N
∑
2 j)1,j*i
u(|ri - rj|) + w(|R - ri|)
(2.28)
The nonequilibrium distribution function Fb(t) for the bath in the presence of a fixed Brownian particle can be derived using
One Brownian Particle in a Nonequilibrium Bath
J. Phys. Chem., Vol. 100, No. 49, 1996 19037
the projection operators Q†2(t) and P†2(t):1
Q†2(t) D(r) ≡ (1 - P†2(t))D(r)
We now rewrite Q†2(y)L′σ(y) as
(2.29)
Q†2(y) L′σ(y) ) Q†2(y)[(L′A(r))*φ(r,y)σ(y)]
P†2(t) D(r) ≡ Tr[D(r) C(rγ)]*〈CC〉t-1(rγ,rβ)*C(rβ) σ(t)
) -[Q2(y)A˙ (r)]*φ(r,y) σ(y) [Q2(y) vB(R)‚∇R[A(r)*φ(r,y)]]σ(y)
(2.30) We will also make use of the projection operator P2(t) which is defined by
P2(t) D(r) ≡ 〈D(r) C(rγ)〉t*〈CC〉t-1 (rγ,rβ)*C(rβ)
) -[Q2(y)N˙ (r)]*φN(r,y) σ(y) [Q2(y) P˙ (r)]*φP(r,y) σ(y) [Q2(y) E˙ (r)]*φE(r,y) σ(y) [Q2(y)vB(R)‚∇R[E(r)]*φE(r,y)]σ(y) (2.42)
(2.31)
The projection operator P†2(t) has the properties where
P†2(t) Fb(t) ) σ(t)
(2.32) N˙ (r) ) -∇r‚(P(r)/m)
(2.43)
and N
P†2(t)
F˙b(t) ) σ˘ (t)
(2.33)
P4 (r) ) -∇r‚τ(r) - ∑ ∇riVδ(r - ri)
(2.44)
E˙ (r) ) -∇r‚JE(r)
(2.45)
i)1
These properties follow from the fact that the thermodynamic forces were selected in such a way that exact value of the average of the dynamical variables C can be obtained from the local equilibrium distribution function σ(t):
C(r)(t) ≡ Tr[Fb(t) C(r)] ) Tr[σ(t) C(r)] ≡ 〈C(r)〉t
(2.34)
The time derivative of Fb(t) is given by
and ∇riV is the force on the Brownian particle exerted by the ith bath particle. This term is present in the expression for P˙ (r) because the momentum density of the bath is not conserved in the presence of a fixed Brownian particle. The quantities τ(r) and JE(r) are the microscopic stress tensor and energy current, respectively. They are given by
∂Fb(t)/∂t) L′Fb(t) N
) L′(P†2(t) + Q†2(t))Fb(t) ) L′(σ(t) + χ(t))
(2.35)
j)1
[
pj‚pj
-
m
1 2
]
∑l rjl∇r u(|rj - rl|) δ(r - rj) j
(2.46)
and
where
P†2(t) Fb(t) ) σ(t) Q†2(t)
Fb(t) ) χ(t)
(2.36) (2.37)
and
N
JE(r) ) ∑ j)1
[
ejpj
-
m
(2.38)
Applying the projection operator Q†2(t) to eq 2.35, we obtain
∂Fb(t) ∂χ(t) ) ) Q†2(t)(L′σ(t)) + Q†2(t)(L′χ(t)) ∂t ∂t
1
t
t
t † y 2
† Q†2(y) L′σ(y) ) +∇r‚JED (r)*φE(r,y) σ(y) +
t
†
Fb(t) ) σ(t) + T+e∫0Q2(s)L′dsχ(0) + † (r))*φE(r,y) σ(y) dy + ∫0t T+e∫ Q (s)L′ds(∇r‚JED t ∫0 T+e∫ Q (s)L′ds[∇r‚τD† (r)*φP(r,y) + t † y 2
t † y 2
∇r[v‚τD† (r)]*φE(r,y)]σ(y) dy (2.49)
χ(0) +
∫0 T+e∫ Q (s)L′ds Q†2(y) L′σ(y) dy t
In the reduced momentum notation, Q†2(y)L′σ(y) becomes
†
Substituting eq 2.40 into eq 2.38, we obtain the following exact expression for Fb(t):
Fb(t) ) σ(t) + T+e
]
(rj - R)pj∇rjw(|R - rj|) δ(r - rj) (2.47)
where J†E(r) is the τ†(r) are given by eqs 2.47 and 2.46 with the ej and pj replaced by their dagger counterparts. We can now rewrite Fb(t) as
(2.40)
t † ∫0Q2(s)L′ds
j
(2.39)
χ(t) ) T+e∫0Q2(s)L′ds χ(0) + ∫0 T+e∫yQ2(s)L′ds Q†2(y)L′σ(y) dy †
∑l rjlpj∇r u(|rj - rl|) -
[∇r‚τD† (r)*φP(r,y) + ∇r[v‚τD† (r)]*φE(r,y)]σ(y) (2.48)
The formal solution to this equation is t
1 2m
m σ(t) + χ(t) ) Fb(t)
Q†2(t)
τ(r) ) ∑
(2.41)
This form for Fb(t) will be used to derive the Fokker-Planck equation.
19038 J. Phys. Chem., Vol. 100, No. 49, 1996
Shea and Oppenheim
We note that the choice of L′ as the Liouvillian governing the time evolution of Fb(t) ensures that L′σ(t) ) 0 in an uniformly flowing system. This imposes the hydrodynamic boundary condition at the surface of the Brownian particle that
vB(R) ) v(R)
(2.50)
These velocities are taken to be small. 2.3. Derivation of the Fokker-Planck Equation. We shall use the projection operator P1(t) to derive an expression for the reduced nonequilibrium distribution function W(XB,t). P1(t) is defined by its action on an arbitrary dynamical variable D(r) by2,3
P1(t) D(r) ) Fb(t)Tr[D(r)]
to rewrite eq 2.58 as
z˘ (t) ) Q1(t)[Lz(t)] + Q1(t)[LB†(Fb(t) W(t))] The formal solution of z(t) is given by t
z(t) ) T+e∫0dsQ1(s)Lz(0) + ∫0 dτ T+e∫τdsQ1(s)LQ1(τ)LB†(Fb(τ) W(τ)) (2.61) t
t
Substituting eq 2.61 in eq 2.57, we obtain the exact master equation for W(t)
[(
]
)
PB*† + vB(R) ‚∇R + ∇P*B†‚〈∇RV〉t W(t) + m
W˙ (t) ) -
∇P* †‚∫ dy Tr[∇RVT+e 0 B
t † ∫0dsQ2(s)L′
t
(2.51)
∇P*B†‚∫0 dy Tr[∇RVT+e
t † ∫ydsQ2(s)L′
t
Using eq 2.15, we find that W˙ (t) is given by
χ(0)]W(t) +
† (∇r‚JED (r))*φE(r,y) ×
∫ydsQ2(s)L′ × σ(y) W(t)] + ∇P* †‚∫ dy Tr[∇RVT+e 0 B t
t
W˙ (t) ) Tr[L(P1(t) + Q1(t))F(t)] ) Tr[L(y(t) + z(t))] (2.52)
(2.60)
†
[∇r‚τD† (r)*φP(r,y) + ∇r[v‚τD† (r)]*φE(r,y)]σ(y) W(t)] + ∇P*B†‚Tr[∇RVT+e∫0dsQ1(s)Lz(0)] + 2∇P*B†‚∫0 dτ × t
where
t
t
y(t) ) P1(t) F(t) ) Fb(t) W(t)
Tr[∇RVT+e∫τdsQ1(s)LQ1(τ)LB†(Fb(τ) W(τ))] (2.62)
(2.53)
and
z(t) ) Q1(t) F(t)
(2.54)
We make use of the following properties
Tr[L′D] ) vB(R)‚∇RTr[D]
(2.55)
† P* B ‚∇ Tr[D(r)] + ∇P* †‚Tr[∇RVD(r)] B m R
Tr[LB†D(r)] ) -
We will use the following approximations to obtain a more tractable Fokker-Planck equation: 1. We will choose the initial bath distribution Fb(t) to be of the local equilibrium form σ(t). This simplifies our calculations by making χ(0) ) 0. We note that even if this condition does not hold, the term containing χ(0) is negligible since it decays to zero on a molecular time scale. 2. We will use the fact that the forces φ(r,t) vary slowly in space to approximate the local equilibrium average by a homogeneous local equilibrium average.1,5,6 We introduce the small variable λ. The local equilibrium of an arbitrary variable D is given by
( (
(2.56) to rewrite eq 2.52 as
[(
)
t † ∫0dsQ2(s)L′
∇P*B†‚Tr[∇RVT+e ∇P* †‚∫ dy Tr[∇RVT+e 0 B t
]
† P* B + vB(R) ‚∇R + ∇P*B†〈∇RV〉t W(t) + m
W˙ (t) ) -
t † ∫ydsQ2(s)L′
† (∇r‚JED (r))*φE(r,y) ×
σ(y) W(t)] + ∇P*B†‚ ∫0 dy Tr[∇RVT+e∫ydsQ2(s)L′ × t
(2.63)
We now expand φ(r) in a Taylor series around R and rewrite the A(r)*φ(r,t) term as
χ(0)]W(t) +
t
) )
1 DeA(r)*φ(r,t) N!h3N 〈D〉t ) 1 A(r)*φ(r,t) Tr e N!h3N Tr
N
A(r)*φ(r,t) ) ∑ ai‚φ(R,t) + i)1
†
N
[∇rτD† (r)*φP(r,y) + ∇r[v‚τD† (r)]*φE(r,y)]σ(y) W(t)] + ∇P* †‚Tr[∇RVz(t)] (2.57) B
λ ∑ ai(ri - R)‚∇Rφ(R,t) + ... (2.64) i)1
We can now express 〈D〉t in terms of the homogeneous average:
〈D〉t ) 〈D〉H (R,t) +
We will now obtain the expression for z˘ (t):
N
z˘ (t) )
λ〈D ˆ (∑ ai(ri - R))〉H(R,t)‚∇Rφ(R,t) + ... (2.65)
d [Q (t) F(t)] ) Q1(t)[Ly(t) + Lz(t)] - P˙ 1(t) F(t) dt 1 (2.58)
where
D ˆ ≡ D - 〈D〉H
We make use of the fact that
F˙b(t) ) L′Fb(t)
i)1
(2.59)
(2.66)
3. We will keep terms only up to quadratic order in and λ. Up to this order, the initial term containing z(0) is negligible
One Brownian Particle in a Nonequilibrium Bath
J. Phys. Chem., Vol. 100, No. 49, 1996 19039
and the upper limit of the time integrals over dτ can be extended from t to ∞.2-4 This follows from the fact that eL′tz(0) and the correlation functions decay to zero for t > τb, where τb corresponds to the relaxation time of the isolated bath. 4. We will make use of the boundary condition vB(R) ) v(R). Using these approximations, eq 2.62 becomes
W˙ (R,P,t) ) -
PB ‚∇RW(t) + M
N
h ∇R(P)(R,t)∇PBW(t) + λ ∑ 〈∇RV(ri - R)〉H:βV
The last term contains the usual dissipative terms found in the Fokker-Planck equation. The velocity term in this last expression is a friction terms that reflect the drag induced by the bath particles. For the purpose of estimating the order of magnitude of this term, we shall consider the friction term to follow the Stokes-Einstein law γ ) 6πνσ where ν is the viscosity. As several authors have pointed out,5,12,13 this is not entirely true for nonequilibrium systems. In such systems, the friction coefficient will depend on a number of factors such as the Reynolds number. These factors are however only small corrections to the Stokes-Einstein law and will not affect the overall order of magnitude in σ of the friction term.
i)1
N
h )(ri - R)〉H:∇R(-β)(R,t)∇PBW(t) + λ ∑ 〈∇RV (e†i - H i)1
λ ∫0 dτ ∞
† 〈JEDT e-L′Q2H(t)τ∇RV〉H:∇R(β)(R,t)
[ [
∫0∞ dτ 〈∇RVe-L′τ∇RV〉H:∇PB β(R,t)
PB M
∇PBW(t) +
] ]
- v(R,t) + ∇PB × W(t) (2.67)
where Q2H(t) corresponds to the projection operator Q2(t) with homogeneous local equilibrium averages. We have made use of the thermodynamic relation
∇R(βµ) ) H h ∇R(β) + βV h ∇R(P)
(2.68)
where P is the pressure and V h and H h correspond to the volume and enthalpy per bath particle, respectively.
4. Comparison to Other Work Our Fokker-Planck equation (2.67) agrees with the one derived by Zubarev et al.5 and by Perez-Madrid et al.6 for the case of a Brownian particle in a moving bath in the presence of a temperature gradient. Zubarov et al.5 derived the FokkerPlanck equation using statistical mechanics while Perez-Madrid et al.6 used a nonequilibrium thermodynamic method of internal degrees of freedom. N 〈∇RV (e†i - H h )(ri - R)〉H: The term involving ∑i)1 ∇R(-β)(R,t) which comes from the homogeneous expansion of 〈∇RV〉t does not appear in the Fokker-Planck equation of Perez-Madrid et al. The authors6 attribute the absence of this term to its extreme microscopic nature which their thermodynamic theory cannot account for. This term is present in the Zubarov et al. equation, although not explicitly, since their Fokker-Planck equation is expressed in terms of local inhomogeneous averages.
3. Analysis of the Terms in the Fokker-Planck Equation The Fokker-Planck equation contains Euler terms of order and λ and dissipative terms of orders λ and 2. The first term of eq 2.67 corresponds to the usual streaming term which is present in the Fokker-Planck equation of a Brownian-bath system for which the bath is in equilibrium. The second term is a streaming term due to a pressure gradient. In order to get an estimate of the order of magnitude of the N 〈∇RV(ri - R)〉H which appears in this correlation function ∑i)1 term, we shall model the potential w(r01) by a hard-sphere potential and the potential u(r12) by a square-well potential:
w(r01)) ∞
for r01 < σ
)0
for r01 > σ
(3.1)
where σ is the radius of the Brownian particle plus that of the bath particle and
u(r12)) ∞
for r12 < a
)C
for a < r12 < Ra
)0
for r12 > Ra
(3.2)
The quantity a is the diameter of the bath particle, R is a small positive number, and C is a constant. Detailed calculation are given in the Appendix. We find this term to be of order σ3. The third term is a streaming term due to a temperature N 〈∇RV e†l (rl - R)〉H‚ gradient. The correlation function ∑l)1 ∇R( - β) is examined in the Appendix. We find this term to be of order σ3. The fourth term is a dissipative term corresponding to the heat flow resulting from a gradient in the temperature.
5. Generalized Langevin Equation for an Arbitrary Function of the Position and Momentum of the Brownian Particle 5.1. Derivation of the Langevin Equation. We shall use the Hermitian adjoint of the projection operator P1(t) to derive the Langevin equation for an arbitrary function G(R,P). P†1(t) is defined by its action on an arbitrary dynamical variable D(r) by
P†1(t) D(r) ) Tr[D(r) Fb(t)]
(5.1)
The time evolution of G˙ (R,P) is given by
G˙ (R,P,t) ) e-LtG˙ (R,P) ) -e-Lt(P†1(t) + Q†1(t))LBG(R,P) PB ) -e-LtP†1(t) ‚∇RG + ∇RV‚∇PBG M e-LtQ†1(t)∇RV‚∇PBG) (5.2)
(
)
The evolution operator e-Lt is given by1 e-Lt ) e-LtP†1(t) -
∫ ds e t
0
t † ∫0dτ-LQ1(τ)
Q†1(0)T-e
t
†
-Ls † P1(s)LQ†1(s)T-e∫sdτ-LQ1(τ)Q†1(t)
Q†1(t) -
∫ ds e t
0
t † ∫sdτ-LQ1(τ)
-Ls † P˙ 1(s)T-e
+
(5.3)
Introducing the small parameter and substituting the expression for e-Lt into the second term of eq 5.2, we obtain the following expression for G˙ (R,P,t):
19040 J. Phys. Chem., Vol. 100, No. 49, 1996
((
)
† P* B + vB(R) ‚∇RG + m
G˙ (R,P,t) ) -e-LtP†1(t) -
)
-Lt
G˙ (R,P,t) ) -K(t,0) + e
-Ls † † P1(s)LK(t,s) + ∇RV‚∇P* B G - K(t,0) + ∫0 ds e t
∫0 ds e-Ls P˙ †1(s) K(t,s) + ∫0 ds e-Ls P˙ †1(s) × t
† P* B
m
]
+ vB(R) ‚∇RG -
† h ∇R(P)(R,t)∇P* λ ∑ e-Lt 〈∇RV(ri - R)〉H:βV B Gi)1
t
t
N
[
Shea and Oppenheim
N
†
P†1(s)T-e∫sdτ-LQ1(τ)Q†1(t)∇RV (5.4)
† h )(ri - R)〉H:∇R(-β)(R,t)∇P* λ ∑ e-Lt 〈∇RV (e†i - H B Gi)1
† † λ ∫0 dτ e-Lt〈JEDT e-L′Q2H(t)τ∇RV〉H:∇R(β)(R,t)∇P* B G∞
where we have defined the fluctuating force K(t,s):
K(t,s) )
t
[
2 ∫0 dτ e-L(t-τ) β(R,t) ∞
†
Q†1(s)T-e∫sdτ-LQ1(τ)Q†1(t)∇RV‚∇PB*†G
(5.5)
P†1(s) K(t,s) ) 0
(5.6)
The second term is at least of order and can be neglected. We now rewrite the Langevin equation as
†
) e-Lt〈∇RV〉t + e-LtTr[∇RVT+e∫0dsQ2(s)L′χ(0)] + e
σ(y)] + ∫0 dy Tr[∇RVT+e † -Lt t ∫ dsQ (s)L′ e ∫0 dy Tr[∇RVT+e [∇rτD(r)*φP(r,y) +
-Lt
t
t
†
∫ydsQ2(s)L′ t y
† (∇r‚JED (r))*φE(r,y)
† 2
∇r[v‚τD† (r)]*φE(r,y)]σ(y)] (5.7) 2.
+∫0 ds e-Ls P†1(s)LK(t,s) ) t
G˙ (R,P,t) ) -K′(t,0) +
N
h ∇R(P )(R,t)(∇PBG)(t) - λ ∑ 〈∇RV (e†i - H h )× R)〉H:βV i)1
(ri - R)〉H:∇R(-β)(R,t)(∇PBG)(t) - λ ∫0 dτ × ∞
([ [
〈∇RVe-L′τ∇RV〉H: β(R,t)
2 ∫0 ds e-Ls P†1(s)LB†K(t,s) (5.8) t
3.
+∫0 ds e
P˙ †1(s)
K(t,s) ) - ∫0 ds e-Ls P†1(s)L′K(t,s) (5.9)
where we have made use of the fact that F˙b(s) ) L′Fb(s)
+∫0 ds e-Ls P˙ †1(s) P1(s)T-e∫sdτ-LQ1(τ)Q1(t)∇RV ) 0 (5.10) t
t
†
we rewrite G˙ (t) as
[
]
† P* B + vB(R) ‚∇RG m
G˙ (R,P,t) ) -K(t,0) + e-Lt
t † ∫0dsQ2(s)L′
† -Lt Tr[∇RVT+e e-Lt〈∇RV〉t‚∇P* B G - e
t
×
†
† (∇r‚JED (r))*φE(r,y) σ(y)‚∇PB*†G] (5.11)
+ e-Lt∫0 dy Tr[∇RVT+e t
t † ∫ydsQ2(s)L′
[∇rτD† (r)*φP(r,y) +
† ∇r[v‚τD† (r)]*φE(r,y)]σ(y)]‚∇P* B G] + t 2 0 ds e-Ls P†1(s)LB†K(t,s)
∫
M
] ]
- v(R,t) - ∇PB ∇PBG(t)
t
(5.15)
†
K′(t,0) ) Q†1(0)T-e∫0dτ-LQ1(τ) Q†1(t)∇RV‚∇PBG
(5.16)
5.2. Average Langevin Equation. The Fokker-Planck and the Langevin equation derived in sections 2 and 5 clearly yield identical average Langevin equations for an arbitrary function G(R,P). The following averge Langevin equation is obtained from the Fokker-Planck equation by averaging G(R,P) over W(t). Note that the random force term K is not present in this equation since the derivation of the Fokker-Planck equation involved an integration over the bath.
〈G˙ (R,P,t)〉 ) +
† -Lt χ(0)‚∇P* ∫0 dy Tr[∇RVT+ e∫ydsQ2(s)L′ × B G] - e t
PB
where K′(t,0) is given by
t
4.
)
∞
∫0 ds e-Ls P†1(s)L′K(t,s) +
-Ls
(
N PB ‚∇RG (t) - λ ∑ 〈∇RV(ri M i)1
† e-L′Q2H(t)τ ∇RV〉H:∇R(β)(R,t)(∇PBG)(t) - ∫0 dτ × 〈JEDT
t
t
† ∇RV〉H‚∇P* B G (5.13)
t
Tr[∇RVFb(t)] t
]
e-L(t-τ) f (R,P) ) f (R,P,t) - ∫t-τ f˙ (R,P,s) ds (5.14)
-Lt
) e
-L′τ
- ∇P*B† ‚
Let us look more closely at the terms of order 2. These terms contain the expression e-L(t-τ) f (R,P) which can be rewritten as
Making use of the following identities:
1.
m
〈∇RVe
The fluctuating force has the property that
e-LtP†1(t)∇RV
P* B
〈(
)〉
PB ‚∇RG (t) m
N
h ∇R(P )(R,t)(∇PBG)(t)〉 λ ∑ 〈〈∇RV(ri - R)〉H:βV i)1
N
h )(ri - R)〉H:∇R(-β)(R,t)(∇PBG)(t)〉 λ ∑ 〈〈∇RV (e†i - H i)1
† λ ∫0 dτ 〈〈JEDT e-L′Q2H(t)τ∇RV〉H:∇R(-β)(R,t)(∇PBG)(t)〉 ∞
(5.12)
Using the same approximations as for the Fokker-Planck equation, eq 3.11 becomes
∫0∞ dτ
〈
([ [
〈∇RVe-L′τ∇RV〉H‚ β(R,t)
] ] )〉
PB M
- v(R,t) -
∇PB ‚∇PBG (t) (5.17)
One Brownian Particle in a Nonequilibrium Bath
J. Phys. Chem., Vol. 100, No. 49, 1996 19041
6. Conclusion In this paper, we have dealt with the problem of Brownian motion in a nonequilibrium bath in the most general and rigorous manner possible. We have described the bath by the exact nonequilibrium distribution function of Levin and Oppenheim,1 which is valid for systems nonlinearly displaced from equilibrium. We started with the Hamiltonian equations for a Brownian particle interacting with N light particles and proceeded to derive a Fokker-Planck equation and a generalized Langevin equation for an arbitrary function G(R,P) using two different time-dependent projection operators. The FokkerPlanck equation was obtained by projecting out the bath variables, while the derivation of the Langevin equation involved an averaging over the nonequilibrium distribution function of the bath. The two equations obtained are equivalent. The Fokker-Planck and Langevin equations were expressed in terms of correlation functions over homogeneous local equilibrium averages. These equations are valid up to second order in the small parameters and λ. The Fokker-Planck equation contains the usual Euler and dissipative equilibrium terms, as well as a number of terms due to the nonequilibrium nature of the bath. Streaming and dissipative terms reflecting spatial variations in pressure, velocity, and temperature are present in these equations. We intend in future work to extend our treatment to a system consisting of n Brownian particles. We hope to use this as a model for two-phase flow and to derive the hydrodynamic equations pertaining to this system. Acknowledgment. J.-E.S. thanks the Natural Sciences and Engineering Council of Canada (NSERC) and the Fonds pour la Formation de Chercheurs et l’Aide a` la Recherche (FCAR) for support.
We now integrate over the spatial argument and the angles to obtain
〈F01r01〉H ) -
N
N
N
∑ 〈∇RV(ri - R)〉H ) ∑ ∑ 〈Fi0rj0〉H i)1
N2 1 2 3 3 3 σ a (R + e-βC(1 - R3)) + 〈F02r02〉H ) 8π 2 V β 3 1 2 4 4 1 σ a (R + e-βC(1 - R4)) - a6(R6 + e-βC(1 - R6)) 2 24 (7.6)
(
)
Let us look at the third term in more detail. N
〈∇RVel†(rl - R)〉H‚∇R(-β) ∑ i)1 is given by N
N
〈∇RVel†(rl - R)〉H‚∇R(-β) ) ∑ 〈F0jel†(r0l)〉H ∑ l)1 l,j)1 )
∑ Nl,j)1
〈( F0j
∑ j,l
1.
1 1 + w0l + ∑ uij (r0l) ‚∇R(-β)(R,t) 2m 2 i)j 2 H (7.7)
〈
〉
pl†‚pl† F0j (r0l) 2m
N 〈F01r01〉H ) ∫ dr1 F01r01F(1) V N ) ∫ dr1 F01r01e-βV(r01)e-βz(r01) V
(7.2)
( β1) drd e 1 ) (- ) δ(r - σ) β
2
(R,t) (7.8)
N(N - 1)〈F01w02r02〉H (7.9) where
N 1 4π 3 σ V(σ)e-β(σ) Vβ 3
(7.10)
and
〈F01w02r02〉H ) 0
(7.11)
〈Fojulkr0l〉H ) N(N - 1)〈F01u12r01〉H + ∑ j,l,k
3.
N(N - 1)〈F01u12r02〉H + N(N - 1)(N - 2)〈F01u23r01〉H (7.12)
-βV(r01)
(7.3)
where
〈F01u12r01〉H )
to rewrite eq 7.2 as
〈F01r01〉H ) -
(kBT)
〈F0jw0lr0l〉H ) N 〈F01w01r01〉H + ∑ j,l
01
01
( )
where 〈F01r01〉H and 〈F01r02〉H are given by eq 7.5 and eq 7.6, respectively.
〈F01w01r01〉H ) -
where z(r01) is a potential of mean force. We now make use of the fact that
F01e-βV(r01)) -
) (N 〈F01r01〉H + H
N(N - 1)〈F01r02〉H)
) N〈F01r01〉H + N(N - 1)〈F01r02〉H (7.1) where the subscript 0 stands for the Brownian particle and Fi0 ) [dV(r0i)/dr0i]rˆ01. Let us first consider the term 〈F01r01〉H
) 〉
pl†‚pl†
Let us now evaluate each term in eq 7.7:
2.
i)1 j)1
(7.5)
In a similar manner, we find that
Appendix Let us look more closely at the second term appearing in the Fokker-Planck equation. We can rewrite the correlation N 〈∇RV(ri - R)〉H as function ∑i)1
N 1 4π 3 -βz(σ) Iσ e Vβ 3
NI 1 V3 β
()
∫ dr01 r01 δ(r01 - σ)e-βz(r)
where I is the unit tensor.
(7.4)
and
2 1 N2 8π2 1 ICe-βC σ3a3(R3 - 1) + σ2a4(R4 - 1) (7.13) β V2 3 3 4
[
]
19042 J. Phys. Chem., Vol. 100, No. 49, 1996
Shea and Oppenheim
〈F01u12r02〉H ) -
1 1 N2 2 -βC 2 3 3 8π Ce - σ a (1 - R3) - σ2a4(1 - R4) + β V2 3 2 1 6 a (1 - R6) (7.14) 24
[
]
References and Notes (1) (2) 231. (3) (4) (5)
Oppenheim, I.; Levine, R. D. Physica A 1979, 99, 383. Van Kampen, N. G.; Oppenheim, I. Physica A 1986, 138, 1986 Romero-Rochin, V.; Oppenheim, I. J. Stat. Phys. 1988, 53, 307. Oppenheim, I.; McBride, J. Physica A 1990, 165, 279. Zubarev, D. N.; Bashkirov, A. G. Physica 1968, 39, 334.
(6) 231. (7) (8) 105. (9) (10) (11) (12) 778. (13) (14) (15) (16)
Perez-Madrid, A.; Rubi, J. M.; Mazur, P. Physica A 1994, 212, san Miguel, M.; Sancho, J. M. Physica A 1979, 99, 357. Foister, R. T.; Van de Ven, T. G. M. J. Fluid. Mech. 1980, 96, Rubi, J. M.; Bedeaux, D. J. Stat. Phys. 1988, 53, 125. Godoh, T. J. Stat. Phys. 1900, 59, 371. Miyazaki, K.; Bedeaux, D. Physica A 1995, 217, 53. Perez-Madrid, A.; Rubi, J. M.; Bedeaux, D. Physica A 1990, 163, Bedeaux, D.; Rubi, J. M. Physica A 1987, 144, 285. Mazur, P.; Oppenheim, I. Physica 1970, 50, 241. Kavassalis, T.; Oppenheim, I. Physica A 1988, 148, 521. Schofield, J.; Oppenheim, I. Physica A 1993, 196, 209.
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