Appendix F
F.1 Stochastic Energetics
Consider the reduce Langevin Eq. (2.1),
m
d 2 x
dt 2 = −
dx
dt
+ (2k B T )
1/2 ξ(t)
(F.1)
with = mγ in units of kg/s.
ξ(t) = 0
( F . 2 )
ξ(t 2 )ξ(t 1 ) = δ (t 2 − t 1 )
(F.3)
The Eq. F.1 is an stochastic process, is better written in the way of It´ s calculus
dx =
p
m
dt
(F.4)
dp =
−
p
m
dt + (2Γ k B T )
1/2 dW
(F.5)
where dW = ξ(t)dt, follows the relations of Eqs. F.6, namely
(dW )
2
= dt
dW dt = 0
(dW )
m
= 0, for m > 2
( F . 6 )
We want to obtain the equation for the Kinetic Energy Power,
dE k
dt , namely
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9
175
F.1 Stochastic Energetics
Consider the reduce Langevin Eq. (2.1),
m
d 2 x
dt 2 = −
dx
dt
+ (2k B T )
1/2 ξ(t)
(F.1)
with = mγ in units of kg/s.
ξ(t) = 0
( F . 2 )
ξ(t 2 )ξ(t 1 ) = δ (t 2 − t 1 )
(F.3)
The Eq. F.1 is an stochastic process, is better written in the way of It´ s calculus
dx =
p
m
dt
(F.4)
dp =
−
p
m
dt + (2Γ k B T )
1/2 dW
(F.5)
where dW = ξ(t)dt, follows the relations of Eqs. F.6, namely
(dW )
2
= dt
dW dt = 0
(dW )
m
= 0, for m > 2
( F . 6 )
We want to obtain the equation for the Kinetic Energy Power,
dE k
dt , namely
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9
175