16
2 The Fokker-Planck Equation
a dimensionless similarity variable for generic potentials. In [11] was performed a
numerical solution of the space FFPE. In [12], the globally connected active rotators
with excitatory and inhibitory connections were analyzed using the FPE. In [13]
a FPE with velocity-dependent coefficients was considered for various isotropic
systems on the basis of probability transition approach. In [14] was introduced
a new method for solving the FPE arising in the simulation of dilute polymeric
solutions. In [15] using the Lie algebraic approach was derived the exact diffusion
propagator of the class of FPEs with logarithmic factors in diffusion and drifts
terms. In [16] based on the FPE, a formula for the effective diffusion coefficient was
derived for a Brownian particle undergoing one dimensional motion in a two-state
Brownian motor (ratchet) in which a spatially periodic, asymmetric potential acting
on the particle switches between two states stochastically. In [17] was presented an
analytical solution of the FPE of non-degenerate parametric amplification system for
generation of squeezed light. In [18] was studied the entropy decay of discretized
FPEs. In [19] was studied the influence of the periodic potential structure on the
diffusion mechanism of a Brownian particle using the FPE. In [20] employing the
FPE, was discussed the stability of the self-similar solution on network models.
In [21] was presented a procedure for deriving general nonlinear FPEs directly
from the master equation. In [22] based on the canonical formalism, the dilatation
symmetry is implemented to the FPE for the Wigner distribution that describes
atomic motion in an optical lattice. In [23] was performed a multiple scales analysis
of the FPE for simple shear flow. In [24] using the Hanggi ansatz to truncate the
evolution equation for probability density, an approximate FPE was derived. In [25]
was analyze the long-time behaviour of transport equations for a class of dissipative
quantum systems with Fokker-Planck type diffusion operator, subject to confining
potentials of harmonic oscillator type.
One way cells accomplish biomolecular transport is through the use of motor
proteins, such as cytoplasmic kinesins and dyneins. Using laser trapping techniques,
the biophysical properties of single motor proteins can now be measured. This,
coupled with increased biochemical and structural data for molecular motors, has
sparked a renewed interest in the mathematical modeling of motor protein function.
In general, mechanistic models of energy transduction in motor proteins must be
studied numerically, because analytic solutions to model equations exist only for
very simple systems [26].
Because the great amount of interactions in these systems, vectors and matrixes
oriented methods are needed, suited to work with Matlab/Octave. We present here
the solution of the FPE [27] by the Crank-Nicholson formalism [28]. The von
Neumann amplification factor, ξ (k) [29], is independent of dt, so the method is
stable for any size dt.
2 The Fokker-Planck Equation
a dimensionless similarity variable for generic potentials. In [11] was performed a
numerical solution of the space FFPE. In [12], the globally connected active rotators
with excitatory and inhibitory connections were analyzed using the FPE. In [13]
a FPE with velocity-dependent coefficients was considered for various isotropic
systems on the basis of probability transition approach. In [14] was introduced
a new method for solving the FPE arising in the simulation of dilute polymeric
solutions. In [15] using the Lie algebraic approach was derived the exact diffusion
propagator of the class of FPEs with logarithmic factors in diffusion and drifts
terms. In [16] based on the FPE, a formula for the effective diffusion coefficient was
derived for a Brownian particle undergoing one dimensional motion in a two-state
Brownian motor (ratchet) in which a spatially periodic, asymmetric potential acting
on the particle switches between two states stochastically. In [17] was presented an
analytical solution of the FPE of non-degenerate parametric amplification system for
generation of squeezed light. In [18] was studied the entropy decay of discretized
FPEs. In [19] was studied the influence of the periodic potential structure on the
diffusion mechanism of a Brownian particle using the FPE. In [20] employing the
FPE, was discussed the stability of the self-similar solution on network models.
In [21] was presented a procedure for deriving general nonlinear FPEs directly
from the master equation. In [22] based on the canonical formalism, the dilatation
symmetry is implemented to the FPE for the Wigner distribution that describes
atomic motion in an optical lattice. In [23] was performed a multiple scales analysis
of the FPE for simple shear flow. In [24] using the Hanggi ansatz to truncate the
evolution equation for probability density, an approximate FPE was derived. In [25]
was analyze the long-time behaviour of transport equations for a class of dissipative
quantum systems with Fokker-Planck type diffusion operator, subject to confining
potentials of harmonic oscillator type.
One way cells accomplish biomolecular transport is through the use of motor
proteins, such as cytoplasmic kinesins and dyneins. Using laser trapping techniques,
the biophysical properties of single motor proteins can now be measured. This,
coupled with increased biochemical and structural data for molecular motors, has
sparked a renewed interest in the mathematical modeling of motor protein function.
In general, mechanistic models of energy transduction in motor proteins must be
studied numerically, because analytic solutions to model equations exist only for
very simple systems [26].
Because the great amount of interactions in these systems, vectors and matrixes
oriented methods are needed, suited to work with Matlab/Octave. We present here
the solution of the FPE [27] by the Crank-Nicholson formalism [28]. The von
Neumann amplification factor, ξ (k) [29], is independent of dt, so the method is
stable for any size dt.