1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
13
not bound to be differentiable in time. Only when the operation = d ˜
/dt makes
sense, can one define the derivative of v with respect to time and use (1.15).
Equation (1.21) is called stochastic differential equation. One may formally integrate it, but it yields concrete results only on the statistical properties of the quantities
appearing in it. Therefore, its terms should not be interpreted literally as velocities
or forces; in particular, one should expect m to obey the action-reaction principle,
especially in nonequilbrium condtions, see e.g. [5–8]. Let us integrate it first, with
formal initial condition v(0) = v 0 . One obtains:
v(t) = e
−γt
⎛
⎝ v 0 +
t
0
e
γt
(t
)dt
⎞
⎠ = v 0 e
−γt
+
t
0
e
−γ(t−t
)
(t
)dt
(1.22)
Then, averaging, one can write:
E[v(t)] = v 0 e
−γt
+
t
0
e
−γ(t−t
)
E[(t
)]dt
= v 0 e
−γt
(1.23)
because E[(t
)] = 0. This result means that on average the velocity of suspended
(Brownian) particles follows the classical hydrodynamic law: althugh no particle
slows down, on average their speed decreases exponentially in time.
8 One may then
compute the correlation function, starting from the following formal expression:
v(t 1 )v(t 2 ) =
⎛
⎝ v 0 e
−γt 1 +
t 1
0
e
−γ(t 1 −t
1 )
(t
1 )dt
1
⎞
⎠
⎛
⎝ v 0 e
−γt 2 +
t 2
0
e
−γ(t 2 −t
2 )
(t
2 )dt
2
⎞
⎠
= v
2
0 e
−γ(t 1 +t 2 )
+ v 0
t 2
0
e
−γ(t 1 +t 2 −t
2 )
(t
2 )dt
2 + v 0
t 1
0
e
−γ(t 1 +t 2 −t
1 )
(t
1 )dt
1
+
t 1
0
dt
1
t 2
0
dt
2 e
−γ(t 1 +t 2 −t
1 −t
2 )
(t
1 ))(t
2 )
(1.24)
Averaging again over all realizations of the process, we obtain
E[v(t 1 )v(t 2 )] = v
2
0 e
−γ(t1+t2) + v 0
t1
0
e
−γ(t1+t2−t
1 ) E[(t
1 )]dt
1 + v 0
t2
0
e
−γ(t1+t2−t
2 ) E[(t
2 )]dt
2
8 Note that the averaging operation has been exchanged with the time integral in (1.23).This justified,
since one averages over a large but always finite number of independent particles, hence E[v] can
be understood as (1/N )
N
i=1 v i , where i denotes the ith Brownian particle.
13
not bound to be differentiable in time. Only when the operation = d ˜
/dt makes
sense, can one define the derivative of v with respect to time and use (1.15).
Equation (1.21) is called stochastic differential equation. One may formally integrate it, but it yields concrete results only on the statistical properties of the quantities
appearing in it. Therefore, its terms should not be interpreted literally as velocities
or forces; in particular, one should expect m to obey the action-reaction principle,
especially in nonequilbrium condtions, see e.g. [5–8]. Let us integrate it first, with
formal initial condition v(0) = v 0 . One obtains:
v(t) = e
−γt
⎛
⎝ v 0 +
t
0
e
γt
(t
)dt
⎞
⎠ = v 0 e
−γt
+
t
0
e
−γ(t−t
)
(t
)dt
(1.22)
Then, averaging, one can write:
E[v(t)] = v 0 e
−γt
+
t
0
e
−γ(t−t
)
E[(t
)]dt
= v 0 e
−γt
(1.23)
because E[(t
)] = 0. This result means that on average the velocity of suspended
(Brownian) particles follows the classical hydrodynamic law: althugh no particle
slows down, on average their speed decreases exponentially in time.
8 One may then
compute the correlation function, starting from the following formal expression:
v(t 1 )v(t 2 ) =
⎛
⎝ v 0 e
−γt 1 +
t 1
0
e
−γ(t 1 −t
1 )
(t
1 )dt
1
⎞
⎠
⎛
⎝ v 0 e
−γt 2 +
t 2
0
e
−γ(t 2 −t
2 )
(t
2 )dt
2
⎞
⎠
= v
2
0 e
−γ(t 1 +t 2 )
+ v 0
t 2
0
e
−γ(t 1 +t 2 −t
2 )
(t
2 )dt
2 + v 0
t 1
0
e
−γ(t 1 +t 2 −t
1 )
(t
1 )dt
1
+
t 1
0
dt
1
t 2
0
dt
2 e
−γ(t 1 +t 2 −t
1 −t
2 )
(t
1 ))(t
2 )
(1.24)
Averaging again over all realizations of the process, we obtain
E[v(t 1 )v(t 2 )] = v
2
0 e
−γ(t1+t2) + v 0
t1
0
e
−γ(t1+t2−t
1 ) E[(t
1 )]dt
1 + v 0
t2
0
e
−γ(t1+t2−t
2 ) E[(t
2 )]dt
2
8 Note that the averaging operation has been exchanged with the time integral in (1.23).This justified,
since one averages over a large but always finite number of independent particles, hence E[v] can
be understood as (1/N )
N
i=1 v i , where i denotes the ith Brownian particle.