Chapter 3
Biased Brownian Motion
We use the definition given by Derényi and Astumian [1] for Brownian ratchets,
namely:
“Non-equilibrium fluctuations, whether generated externally or by a chemical reaction far-from-equilibrium, can drive directed motion along an anisotropic structure
without thermal gradients or net macroscopic forces, simply by biasing Brownian
motion. Systems operating on this principle are often referred to as Brownian
ratchets, and the transport in such systems is called fluctuation driven transport”.
For the historical evolution of the fundamental models of ratchet devices see
Chapters 1 and 2 of Cubero and Renzoni [2]. Several references are involved in this
context: Bug and Berne [3], Ajdari and Prost [4], Magnasco [5], Astumian and Bier
[6], Astumian [7], Astumian and Derényi [8].
First we parameterize the Langevin equation, second in order to perform the
numerical simulations we discretize the time, ˜
t and we introduce the “Wiener´ s
increment” dW
˜
t
, see also Appendix E, then we build the Fokker-Plank’s matrices.
Considering a periodic potential slightly tilded we perform simulations using the
Euler’s and Fokker-Plank’s equations. Later we introduce Dichotomous Markov
noise, DMN, and we teach how to generate it. Then we analyze the “Flashing”
Ratchet, by introducing the DMN into the Euler’s equation. Then several simulations are performed with this equation. Finally we introduce the “Rocking” Ratchet
into the Euler’s equation and perform simulations with it.
3.1 Parametrization of the Langevin Equation
We consider a system of Brownian particles of negligible mass, each particle being
acted upon by a force and velocity-dependent friction, i.e. damped motion in the
potential V (x), and also acted by an external force F , in accordance to the Langevin
Eq. (2.2), we have:
© 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_3
25
Biased Brownian Motion
We use the definition given by Derényi and Astumian [1] for Brownian ratchets,
namely:
“Non-equilibrium fluctuations, whether generated externally or by a chemical reaction far-from-equilibrium, can drive directed motion along an anisotropic structure
without thermal gradients or net macroscopic forces, simply by biasing Brownian
motion. Systems operating on this principle are often referred to as Brownian
ratchets, and the transport in such systems is called fluctuation driven transport”.
For the historical evolution of the fundamental models of ratchet devices see
Chapters 1 and 2 of Cubero and Renzoni [2]. Several references are involved in this
context: Bug and Berne [3], Ajdari and Prost [4], Magnasco [5], Astumian and Bier
[6], Astumian [7], Astumian and Derényi [8].
First we parameterize the Langevin equation, second in order to perform the
numerical simulations we discretize the time, ˜
t and we introduce the “Wiener´ s
increment” dW
˜
t
, see also Appendix E, then we build the Fokker-Plank’s matrices.
Considering a periodic potential slightly tilded we perform simulations using the
Euler’s and Fokker-Plank’s equations. Later we introduce Dichotomous Markov
noise, DMN, and we teach how to generate it. Then we analyze the “Flashing”
Ratchet, by introducing the DMN into the Euler’s equation. Then several simulations are performed with this equation. Finally we introduce the “Rocking” Ratchet
into the Euler’s equation and perform simulations with it.
3.1 Parametrization of the Langevin Equation
We consider a system of Brownian particles of negligible mass, each particle being
acted upon by a force and velocity-dependent friction, i.e. damped motion in the
potential V (x), and also acted by an external force F , in accordance to the Langevin
Eq. (2.2), we have:
© 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_3
25