# A shortcut way to the Fokker-Planck equation for the non-Markovian dynamics
: Induced electric field from a time-independent magnetic field

Joydip Das  Department of Chemistry, Visva-Bharati, Santiniketan, India,
731235  Mousumi Biswas  Department of Chemistry, Visva-Bharati, Santiniketan,
India, 731235  Debasish Mondal  Department of Chemistry and Center for
Molecular and Optical Sciences & Technologies, Indian Institute of Technology
Tirupati, Tirupati, India, 517 506  Bidhan Chandra Bag
<EMAIL_ADDRESS>Department of Chemistry, Visva-Bharati,
Santiniketan, India, 731235

###### Abstract

Using the property of the solution of the Langevin dynamics with a generalized
frictional memory kernel and time-dependent deterministic force field (TDFF)
we show that a solution method (which is very simple as well as shortcut) can
be used to derive the Fokker-Planck equation (FPE) for this dynamics. Then the
equation is derived for the non-Markovian dynamics with additional force from
harmonic potential, magnetic fields, and both, respectively. Thus the method
is instructive in deriving the Fokker-Planck equation in a shortcut way in the
presence of an additional time-dependent stochastic force. Here we have to
consider that the relevant drift terms are independent of the random force.
Then another very important point is to be noted here. To interpret FPE, we
recognize that the memory of the non-Markovian dynamics can induce an electric
field from the time-independent magnetic field in the presence of a
conservative force field. The induced electric field may be the underlying
reason for the special kind of correlation between position and velocity for
the non-Markovian dynamics in the presence of a magnetic field. At the same
time, it may include an additional drift term to the memory-influenced
modulation of the effect from the external time-dependent force field. We
understand that the present study, with the recognization of the induced
electric field will bring strong attention to different areas of non-
equilibrium statistical mechanics such as physical tuning of conductivity of
ions in solid electrolytes, noise-induced transition and stochastic
thermodynamics for non-Markovian dynamics etc.

##  I Introduction

The Langevin equation of motion is the expected law of causality for the
Brownian motion. Its equivalent description may be the Fokker-Plank equation
(FPE) for special cases like the Gaussian random force driven Brownian motion.
The FPE equation is unavoidable in calculating the barrier crossing rate
constant1, the entropy production for the irreversible processes3, 4, 5, 6 and
in the context of the linear response theory2 etc. Kramers’ theory 1 on the
barrier crossing dynamics requires the relevant probability density function
(PDF) at the steady-state. To determine this, we must need the FPE1. In this
context, we cite a very recent study, 7 where escape dynamics in a two-
temperature Brownian magneto-system is investigated. Similarly, the Fokker-
Planck operator plays an important role in the linear response theory, as
demonstrated in Ref.2. Thus finding the FPE corresponding to a Langevin
equation and its use of it are always intriguing issues in statistical
mechanics 6, 7, 8. It is to be noted here that although the probability
density function (PDF) is known for the linear stochastic process, still, the
Fokker-Planck equation is required in many cases, as mentioned above. Thus,
one of the objectives of the present study is to derive the Fokker-Planck
equation for the non-Markovian dynamics (of a driven Brownian particle) in a
simple way using the PDF.

The theory for the Markovian dynamics is well established, and its application
is continued, especially to the newly born subject, stochastic thermodynamics.
Now one may cite the famous assertion of van Kampen 9. He argued that the non-
Markovian is the rule, Markovian is an exception 9. Study on the non-Markovian
dynamics over the last four decades in diverse contexts, such as activated
barrier crossing10, 11, 12, 13, 14, stochastic resonance15, ratchet
rectification16, energy harvesting17 and many others18, 19, 20, 21, 22, 23,
24, 25, 4, 26, 27), justifies the assertion. We now note that although the
study on the Markovian dynamics of charged particles started in the early
sixties28 of the last century, the present state of knowledge on the non-
Markovian dynamics of the same in the presence of a magnetic field is at an
early stage20, 21, 22, 29, 30, 31, 23. The reason behind this may be that the
theoretical analysis of the non-Markovian dynamics, in general, is
challenging. For further particulars, we refer to the Refs.9, 32. The
probabilistic description of the non-Markovian dynamics is non-trivial and
highly formidable9, 32. In the theory of Brownian motion, the Langevin and
Fokker- Planck equations correspond to the coarse-grained model, in which only
a few degrees of freedom related to observables of a Brownian particle(BP) are
explicitly retained32. Assuming the environment of a Brownian particle is a
collection of non interecting harmonic oscillators, derivation of the
generalized Langevin equation of motion now becomes a common practice in both
classical and quantum Brownian motions. Recently it has been derived in Ref.33
for a Brownian oscillator in the presence of a fluctuating magnetic field. It
is to be noted here that starting from basic molecular theory (Liouville
equation or the Hamilton’s equations of motion), one may obtain the
generalized Langevin equation of motion using the projection operator
formalism32. Very recently, this formalism has been used in different contexts
34. One may also obtain the Fokker-Planck equation using the projection
operator formalism mago. For the non-Markovian dynamics where the decay of
certain molecular correlation functions has an asymptotic slow inverse power
decay, the projection operator formalism may not be applicable to derive the
Fokker-Planck equation, as a detailed discussion was given in Ref.32. Then the
introduction of additional variables may thus be essential9. Using auxiliary
variables, one may have the continuity equations4, 30, 31, 35, 36, 37, which
is an effective Markovian description of a given non-Markovian design in the
extended phase space. In some cases, such as for a linear system, one may
derive the Fokker-Planck equation for the non-Markovian dynamics without using
any additional variable(s)18, 19, 20, 21, 22, 23, 24. Using the interaction
picture, the Fokker-Planck equations for the non-Markovian Langevin dynamics
with or without isotropic harmonic force field were first derived in 197618.
Twenty years later, a similar issue with a one-dimensional harmonic oscillator
was investigated, introducing a proper characteristic function19. Applying
this approach, the Fokker-Planck equations for the non-Markovian Langevin
dynamics with a magnetic force and a time-dependent force field were addressed
in velocity space and phase space20, respectively. Recently22, it has been
extended to derive the FPE for non-Markovian harmonic oscillator across a
magnetic field and time-dependent force field (TDFF). The revised version of
this Fokker-Planck equation was given in Ref.24. To avoid any confusion, we
would mention here that there is a typo in the same ((Eq.(7)). The quantity,
${\bf{q}}\dot{\nabla}_{{\bf{x}}}P$ in Eq.(7) must be
${\bf{q}}\dot{\nabla}_{{\bf{u}}}P$.

Recently, an alternative solution method has been developed in Ref.21, 23 in
deriving the Fokker-Planck equation for the non-Markovian dynamics. Solving
the given Langevin equation of motion, one may obtain the relevant moments as
well as the probability distribution function. Knowing the nature of these
quantities, a relevant Fokker-Planck equation can be proposed whose
coefficients are to be determined applying two simple mathematical notions,
(1) Setting of linear algebraic equations with the collection of coefficients
of phase space variables and their appropriate multiples after putting the
given distribution function into the proposed Fokker- Planck equation, and (2)
solution of the algebraic equations by the elimination procedure to determine
the unknown coefficients. Thus the method seems to be a simple as well as
shortcut one compared to the other solution-based methods18, 19. It reproduces
all the known results. Here it is to be noted that at the present state of
knowledge, some of the terms in the FPE for the non-Markovian dynamics in the
presence of a magnetic field22, 24, 23 require special attention. In this
context, an essential identification is that the non-Markovian dynamics may
induce an electric field from the time-independent magnetic field in the
presence of a conservative force field. It may be the reason for appearing the
special cross-correlation between the position and the velocity. Thus induced
electric field introduces an additional diffusion term with a cross-
derivative. At the same time, it may modulate other diffusion terms. Thus one
of the key objectives of the present study is to draw special attention to
this physical quantity in the field, non-equilibrium statistical mechanics. In
addition to this, we demonstrate that how the solution method, proposed by das
et. al.21, 23 can be used to determine the modulation of the drift term due to
an external deterministic force field in the presence of a frictional meory
kernel induced velocity dependent feedback. It is to be noted here that the
additional drift term (which is due to the external time-dependent force) in
the proposed Fokker-Planck equation creates a difficulty in using the solution
method 21. The number of independent relations (a set of linear algebraic
equations) among the coefficients which appear in the proposed equation is
less than that of the number of relevant unknowns. In this circumstance, we
need additional conditions based on the physics of the given system.The
solution of the Langevin equation implies that the response function or the
susceptibility does not depend on the external force field. Then we consider
that the drift terms for the other force fields and the relevant diffusion
terms are independent of the external force field. Using this property into
the independent relations we determine the modulation of the drift term due to
an external deterministic force field in the presence of a frictional memory
kernel induced velocity dependent feedback. We show that this technique works
even in the presence of both conservative and non-conservative fields,
respectively. With four examples, we show that the method works well in this
context. This calculation is instructive to determine easily a diffusion term
due to an external stochastic force driven Brownian particle in the presence
of frictional memory induced velocity dependent feedback, harmonic force and
magnetic field. Finally, a reader may find a chronological development of the
relevant Fokker-Planck equations such that it will be helpful to understand
them.

Before leaving this section, we would like to mention the following point.
Consideration of the time-dependent deterministic force field makes the
present study significantly relevant in the field of stochastic
thermodynamics, which is now at an early stage to consider the non-Markovian
dynamics38, 27. The connotation of work in ST demands a time-dependent
deterministic force field 38, 27, 39. We mention a few circumstances in the
closing remarks.

The outlay of the article is as follows. In Sec. II A, we present an
alternative formulation for the derivation of the FPE for the non-Markovian
dynamics in the presence of a time-dependent force field. The Fokker-Planck
equation is derived in the following three consecutive subsections for
additional force from harmonic potential or magnetic field or both of them,
respectively. We conclude the paper in Sec. III.

##  II Memory induced modulation of the effect of a time dependent external
force field

###  II.1 Non-Markovian dynamics of a free Brownian particle in the presence
of a time-dependent force field

The Langevin equation of motion of a free Brownian particle which is coupled
to a non-Markovian thermal bath in the presence of a time-dependent force
field ($\bm{a}(t)$) can be read18, 21 as:

$\dot{\bm{u}}=-\int_{0}^{t}\gamma(t-\tau)\bm{u}(\tau)d\tau+\bm{f}(t)+\bm{a}(t)\;,$
(1)

where $\bm{u}$ is a velocity vector of the particle with mass, $m=1$.
$\bm{f}(t)$, in the above equation is a colored Gaussian thermal noise, and it
is connected to the time-dependent damping strength ($\gamma(t)$) through the
following fluctuation-dissipation relation:

$\langle\bm{f}(t)\cdot\bm{f}(t^{\prime})\rangle=3k_{B}T\gamma(t-t^{\prime})\;.$
(2)

Thus Langevin equation of motion (1) is a special case that is valid only for
stationary noise procees. In other words, the thermal bath is assumed to be in
equilibrium state. It is applicable for other cases also.

We now go through the same steps as in Ref. 21, with the time dependence of
$a$ being the only difference would be needed as clarification. First, we
calculate the relevant moments using the solution of the Langevin Eq.(1). The
solution of this equation can be read as

$\displaystyle\bm{g}(t)$ $\displaystyle=$
$\displaystyle\bm{u}-\chi(t)\bm{u}(0)-\int_{0}^{t}\chi(t-\tau)\bm{a}(\tau)d\tau$
(3) $\displaystyle=$
$\displaystyle\bm{u}-(\bm{c}+\bm{p})=\bm{u}-\bm{c^{\prime}}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\chi(t-\tau)\bm{f}(\tau)d\tau\;\;\;.$

where we have used $\bm{c}^{\prime}=\bm{c}+\bm{p}$ , $\bm{c}=\chi(t)\bm{u}(0)$
and $\bm{p}=\int_{0}^{t}\chi(t-\tau)\bm{a}(\tau)d\tau$. $\chi(t)$, in the
above equation is defined as

$\displaystyle\chi(t)$ $\displaystyle=$
$\displaystyle\mathcal{L}^{-1}[\tilde{\chi}(z)]$ (4) $\displaystyle=$
$\displaystyle\mathcal{L}^{-1}\left[\frac{1}{z+\tilde{\gamma}(z)}\right]\;\;\;.$

Here $\mathcal{L}^{-1}$ denotes Laplace inversion and $\tilde{\gamma}(z)$ is
the Laplace transform of $\gamma(t)$

$\tilde{\gamma}(z)=\int_{0}^{t}e^{-zt}{\gamma}(t)dt\;\;\;.$ (5)

$chi(t)$ in Eqs.(3) is known as the response function or the susceptibility.
One may notice here that it is independent of the external force field.
Shortly we will find that this notice may help to derive the relevant Fokker-
Planck equation in a shortcut way.

All the second moments corresponding to the fluctuations (as given by Eq.(3)
can be represented by the matrix, $\bm{\mathcal{A}}(t)$ with $A_{ij}=\langle
g_{i}(t)g_{j}(t)\rangle$. Now using Eq.(3) ,we have
$\bm{\mathcal{A}(t)}=A(t)I$, where $I$ corresponds to the relevant identity
matrix and

$\displaystyle A(t)$ $\displaystyle=$
$\displaystyle\left(\frac{3k_{B}T}{m}\right)\left[1-\chi^{2}(t)\right]\;\;\;.$
(6)

Thus for the linear Langevin equation (1) with the Gaussian noise, the
velocity distribution function can be written as 2, 18, 21

$P(\bm{u},\bm{u(0)};t)=\left[\frac{3}{2\pi|\bm{\mathcal{A}(t)}|}\right]^{\frac{3}{2}}\exp\left[-\frac{3}{2}\bm{g}{\dagger}(t)\bm{\mathcal{A}}^{-1}(t)\bm{g}(t)\right]\;\;.$
(7)

In the next step, keeping in mind the nature of the distribution function as
well as the matrix, $\bm{\mathcal{A}}(t)$, we may propose the relevant Fokker-
Planck equation as 21

$\frac{\partial P}{\partial{t}}=\bm{G(t)}\cdot\nabla
P+\beta(t)\nabla\cdot\bm{u}P+H(t)\nabla^{2}P\;\;,$ (8)

where $\beta(t)$, $\bm{G(t)}$ and $H(t)$ are time-dependent quantities to
account for the NMD properly. The first term on the right-hand side of the
above equation corresponds to the drift term due to the deterministic time-
dependent force field, $\bm{a}(t)$. The remaining second and last terms are
usual drift and diffusion terms, respectively. Now we have to determine the
time-dependent coefficients, making use of the solution of the above equation.
Using the distribution function (7) in Eq.(8), we obtain the following set of
linear algebraic equations with the collection of coefficients of phase space
variables and their appropriate multiples 21, 23, 40

$\bm{G(t)}-\beta(t){\bm{c^{\prime}}}+\frac{2H(t)\bm{{c^{\prime}}}}{A(t)}=\frac{\dot{A}(t)\bm{{c^{\prime}}}}{A(t)}-\dot{\bm{c^{\prime}}}\;\;\;,$
(9)

and

$2H(t)-2\beta(t)A(t)=\dot{A}(t)\;\;\;.$ (10)

Using Eq.(10) into Eq.(9) we have

$\beta(t)\bm{c^{\prime}}+\bm{G(t)}=-\dot{\bm{c^{\prime}}}\;\;\;.$ (11)

noindent It is to be noted here that the additional drift term
($\bm{G(t)}\cdot\nabla P$) in the proposed Fokker-Planck equation ( which is
due to the external time-dependent force) creates a difficulty to use the
solution method 21. Here we find two linear algebraic equations (9-10) with
the three unknown coefficients, $\beta(t),G(t)$ and $H(t)$, respectively. In
this circumstance, we need additional conditions based on the physics of the
given system. The solution of the Langevin equation implies that the response
function or the susceptibility does not depend on the external force field.
Then we consider that the drift terms for the other force field and the
relevant diffusion term are independent of the external force field. In the
absence of the time-dependent force, $\bm{G(t)}=0$18, 21,
$\bm{c^{\prime}}=\bm{c}$ and then the above equation becomes

$\beta(t)=-\frac{\dot{\chi(t)}}{\chi(t)}\;\;\;,$ (12)

and $H(t)$ is given by

$H(t)=\frac{1}{6}\chi^{2}(t)\frac{d}{dt}\left[\chi^{-2}(t)A(t)\right]\;\;\;.$
(13)

We note that the coefficient, $\beta(t)$ of the drift term due to dissipative
action, is independent of $\bm{c}(t)$ , which (is the average velocity) may be
arbitrary as it depends on the initial condition, $\bm{u}(0)$. In other words,
$\beta(t)$ only depends on the response function as expected. Similarly,
$A(t)$, as well as the diffusion coefficient, $H(t)$, are also independent of
$\bm{c}(t)$ since the dispersion of velocity does not depend on the average
velocity. Thus $\beta(t)$ and $H(t)$ may remain the same in the presence of
time-dependent force field (which changes only the average velocity) since the
response function and the variance, $A(t)$, do not depend on it. Then from
Eq.(11) we have

$\bm{G(t)}=-(\dot{\bm{p}}+\beta(t)\bm{p})\;\;\;,$ (14)

Thus, the Fokker-Planck equation corresponding to the Langevin Eq.(1) is:

$\frac{\partial
P}{\partial{t}}=-\frac{\dot{\chi}(t)}{\chi(t)}\nabla\cdot\bm{u}P-(\dot{\bm{p}}+\beta(t)\bm{p})\cdot\nabla
P+\frac{1}{6}\chi^{2}(t)\frac{d}{dt}\left[\chi^{-2}(t)A(t)\right]\nabla^{2}P\;\;\;.$
(15)

Now, one may check that the distribution function (7) is the solution of the
above equation. It constitutes the necessary and sufficient check of the above
calculation. The above equation was derived earlier in Ref.20 using the
characteristic function. Then one may easily notice the simplicity and the
shortcuts of the present method.

For further check, one can show that the above equation reduces to the
standard result18, 21 in the absence of the time-dependent force field.
Finally, at the Markovian limit,
$\gamma(t-t^{\prime})=2\gamma_{0}\delta(t-t^{\prime})$ , the response function
can be read as, $\chi(t)=\exp^{-(\gamma_{0}(t))}$, where $\gamma_{0}$ is the
damping strength. Then we have
$\beta(t)=\gamma_{0},\dot{\bm{p}}=-\gamma_{0}\bm{p}+\bm{a},H(t)=\gamma_{0}k_{B}T$
and $\bm{G(t)}=-\bm{a}$. Thus at the Markovian limit, Eq.(15) reduces to the
known result 32, 2, 41, 42, 43, 44,

$\frac{\partial
P(\bm{u},t)}{\partial{t}}=\gamma_{0}\nabla\cdot\bm{u}P-\bm{a}\cdot\nabla
P+\gamma_{0}k_{B}T\nabla^{2}P\;\;\;.$ (16)

Comparing this equation with that of Eq.(15), one may uncover in a simple way
how the memory effect (from the time-dependent damping strength) can modulate
the influence of the time-dependent external force field.

###  II.2 Non-Markovian dynamics of time-dependent force driven harmonic
oscillator

For an isotropic harmonic oscillator having a frequency $\omega$, Eq.(1)
becomes

$\ddot{\bm{x}}+\omega^{2}\bm{x}(t)+\int_{0}^{t}\gamma(t-\tau)\dot{\bm{x}}(\tau)d\tau=\bm{f}(t)+\bm{a}(t)\;\;\;.$
(17)

Now following the previous subsection, the probability density function for
the Langevin equation of motion [Eq.(17)] can be written as 2, 18, 21

$P(\bm{x},\bm{u},\bm{x}(0),\bm{u}(0);t)=\left(\frac{1}{2\pi}\right)^{3}\left[\frac{1}{|\bm{\mathcal{A}(t)}|}\right]^{\frac{3}{2}}\exp\left[-\frac{3}{2}\bm{g}{\dagger}(t)\bm{\mathcal{A}(t)}^{-1}\bm{g}(t)\right]$
(18)

with

$\bm{g}(t)=\begin{bmatrix}\bm{g_{1}}(t)\\\ \bm{g_{2}}(t)\end{bmatrix}\;\;,$
(19)

where

$\displaystyle\bm{g_{1}}(t)$ $\displaystyle=$
$\displaystyle\bm{x}(t)-\left[\chi_{\bm{x}}(t)\bm{x}_{0}+\chi_{\bm{u}}(t)\bm{u}_{0}-\int_{0}^{t}\chi_{\bm{u}}(\tau)\bm{a}(t-\tau)d\tau\right]$
(20) $\displaystyle=$
$\displaystyle\bm{x}(t)-\left[\bm{c_{1}}+\bm{q_{x}}\right]$ $\displaystyle=$
$\displaystyle\bm{x}(t)-\bm{{c_{1}}^{\prime}}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\chi_{\bm{u}}(\tau)\bm{f}(t-\tau)d\tau\;\;.$

From the above equation, the fluctuations in velocity can be read as

$\displaystyle\bm{g_{2}}(t)$ $\displaystyle=$
$\displaystyle\bm{u}(t)-\left[\dot{\chi}_{\bm{x}}(t)\bm{x}_{0}+\dot{\chi}_{\bm{u}}(t)\bm{u}_{0}-\int_{0}^{t}\dot{\chi}_{\bm{u}}(\tau)\bm{a}(t-\tau)d\tau\right]$
(21) $\displaystyle=$ $\displaystyle\bm{u}(t)-[\dot{\bm{c_{1}}}+\bm{p_{x}}]$
$\displaystyle=$ $\displaystyle\bm{u}(t)-\bm{{c_{2}}^{\prime}}$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\dot{\chi}_{\bm{u}}(\tau)\bm{f}(t-\tau)d\tau\;\;.$

Here we have used
$\bm{c_{1}}=\left[\chi_{\bm{x}}(t)\bm{x}_{0}+\chi_{\bm{u}}(t)\bm{u}_{0}\right]$,
$\bm{q_{x}}=\int_{0}^{t}\chi_{\bm{u}}(\tau)\bm{a}(t-\tau)d\tau$ and
$\bm{\dot{q_{x}}}=\bm{p_{x}}$. $\chi_{\bm{x}}(t)$ and $\chi_{\bm{u}}(t)$ in
Eq.(20) are the inverse Laplace transform of the relations,
$\tilde{\chi}_{\bm{x}}(s)=\frac{\tilde{\gamma}(s)+s}{s^{2}+s\tilde{\gamma}(s)+\omega^{2}}$
and $\tilde{\chi}_{\bm{u}}(s)=\frac{1}{s^{2}+s\tilde{\gamma}(s)+\omega^{2}}$,
respectively. Then we define the matrix, $\bm{\mathcal{A}(t)}$ with
$A_{ij}=\langle\bm{g_{i}}(t)\cdot\bm{g_{j}}(t)\rangle$.

Now the nature of the distribution function leads to propose the following
Fokker-Planck equation 21, 23, 40 as an equivalent description of Eq.(17),

$\displaystyle\frac{\partial{P}}{\partial{t}}$ $\displaystyle=$
$\displaystyle-\bm{u}\cdot\nabla_{{\bf{x}}}P+\bm{G}(t).\nabla_{{\bf{u}}}P+H_{1}(t)\bm{x}\cdot\nabla_{{\bf{u}}}+H_{2}(t)\nabla_{{\bf{u}}}.{\bf{u}}P$
(22) $\displaystyle+$ $\displaystyle
H_{3}(t)\nabla_{{\bf{u}}}.\nabla_{{\bf{x}}}P+H_{4}(t){\nabla_{{\bf{u}}}}^{2}P\;\;\;,$

where $\bm{G}(t),H_{1}(t),H_{2}(t),H_{3}(t)$ and $H_{4}(t)$ are the relevant
time-dependent coefficients to account for the NMD properly. The first term on
the right-hand side of the above equation is the usual drift term for both
Markovian and non-Markovian dynamics, respectively. In other words, here we
consider that the time-dependent external force field can not change the
definition of velocity. The next term is due to the time-dependent external
force field. The drift term corresponding to the harmonic force field appears
with the coefficient $H_{1}(t)$. The remaining drift term is due to the
dissipative force. Finally, the last two terms are the relevant diffusion
terms. To avoid any confusion, we would mention here that following Ref.21,
23, 40, the diffusion terms with other second derivatives are not considered
here.

We are now in a position to determine the time-dependent coefficients of the
above equation. Following the previous section, we have

$\bm{G}(t)=-\dot{\bm{p_{x}}}-H_{2}(t)\bm{p_{x}}-H_{1}(t)\bm{q_{x}}\;\;,$ (23)

$H_{1}(t)=\tilde{\omega}^{2}(t)\;\;,$ (24)

$H_{2}(t)=\tilde{\beta}(t)\;\;,$ (25)

$H_{3}(t)=\frac{k_{B}T}{\omega^{2}}\left[\tilde{\omega}^{2}(t)-\omega^{2}\right]\;\;,$
(26)

and

$H_{4}(t)=\frac{k_{B}T}{\omega^{2}}\tilde{\beta}(t)\;\;\;,$ (27)

where we have used

$\tilde{\beta}(t)=-\frac{d\ln\Delta(t)}{dt}\;\;\;,$ (28)

$\tilde{\omega}^{2}(t)=\Delta^{-1}(t)\left[\ddot{\chi}_{\bm{u}}(t)\dot{\chi}_{\bm{x}}(t)-\ddot{\chi}_{\bm{x}}(t)\dot{\chi}_{\bm{u}}(t)\right]$
(29)

and

$\Delta(t)=\left[\dot{\chi}_{\bm{u}}(t)\chi_{\bm{x}}(t)-\dot{\chi}_{\bm{x}}(t)\chi_{\bm{u}}(t)\right]\;\;\;.$
(30)

Thus the required Fokker-Planck equation can be read as

$\displaystyle\frac{\partial{P}}{\partial{t}}$ $\displaystyle=$
$\displaystyle-\bm{u}\cdot\nabla_{{\bf{x}}}P-\left[\dot{\bm{p_{x}}}+\tilde{\beta}(t)\bm{p_{x}}+\tilde{\omega}^{2}(t)\bm{q_{x}}\right].\nabla_{{\bf{u}}}P+\tilde{\omega}^{2}(t)\bm{x}.\nabla_{{\bf{u}}}P$
(31) $\displaystyle+$
$\displaystyle\tilde{\beta}(t)\nabla_{{\bf{u}}}\cdot\bm{u}P+\frac{k_{B}T}{\omega^{2}}\left[\tilde{\omega}^{2}(t)-\omega^{2}\right]\nabla_{{\bf{u}}}.\nabla_{{\bf{x}}}P+\frac{k_{B}T}{\omega^{2}}\tilde{\beta}(t){\nabla_{{\bf{u}}}}^{2}P\;\;\;.$

Now, one can check that the distribution function (18) is a solution of the
above equation. Thus we arrive to the new Fokker-Planck equation with a
shortcut way. Shortly we will compare this equation with its counterpart,i.e.,
the Markovian case.

For further check, one may use the condition, $\bm{a}(t)=0$. Then
$\bm{q_{x}}=\bm{p_{x}}=0$ as well as $\bm{G_{1}}(t)=0$ and the above equation
reduces to the standard result18, 19, 21

$\displaystyle\frac{\partial{P}}{\partial{t}}$ $\displaystyle=$
$\displaystyle-\bm{u}\cdot\nabla_{{\bf{x}}}P+\tilde{\omega}^{2}(t)\bm{x}.\nabla_{{\bf{u}}}P+\tilde{\beta}(t)\nabla_{{\bf{u}}}\cdot\bm{u}P$
(32) $\displaystyle+$
$\displaystyle\frac{k_{B}T}{\omega^{2}}\left[\tilde{\omega}^{2}(t)-\omega^{2}\right]\nabla_{{\bf{u}}}.\nabla_{{\bf{x}}}P+\frac{k_{B}T}{\omega^{2}}\tilde{\beta}(t){\nabla_{{\bf{u}}}}^{2}P\;\;\;.$

At the Markovian limit, Eq.(31) reduces to

$\displaystyle\frac{\partial{P}}{\partial{t}}$ $\displaystyle=$
$\displaystyle-\bm{u}\cdot\nabla_{{\bf{x}}}P+\omega^{2}\bm{x}.\nabla_{{\bf{u}}}P+\gamma_{0}\nabla_{{\bf{u}}}\cdot\bm{u}P-\bm{a(t)}\cdot\nabla_{{\bf{u}}}P+\gamma_{0}k_{B}T{\nabla_{{\bf{u}}}}^{2}P\;\;\;.$
(33)

In the absence of the time-dependent force ($\bm{a(t)}=0$) the above equation
becomes the Kramers’ equation for the isotropic harmonic oscillator1.

Comparing Eq.(32) with Eq.(33) we find that as a signature of the memory
effect induced feedback, the frequency of the harmonic oscillator and the
coefficient of the drift term due to dissipative force become time-dependent
for the non-Markovian dynamics. Then one may compare the coefficients of the
diffusion terms (due to the momentum diffusion) in the respective Fokker-
Planck equations. Here again, it is apparent that the coeffcient for the non-
Markovian dyanmics is time-depenent. Finally, comparing Eq.(31), with Eq.(33)
we find that how the modulation of the effect of time-dependent external force
can be complicated by the conservative and dissipative forces in the presence
of the memory effect induced feedback. Thus the present method is very simple
to use compared to the other methods to identify the modulation. Then one may
expect to apply the present method to identify the modulation of the cyclotron
frequency and its response to the effect of the time-dependent force in the
presence of a non-Markovian thermal bath. The following subsection shows that
one may achieve this result through a simple as well as shortcut way following
subsection (A).

###  II.3 Non-Markovian dynamics of a free particle in the presence of a
constant magnetic field and a time-dependent force field

In the presence of a magnetic field, the equation of motion (1) becomes 20:

$\dot{\bm{u}}=\bm{F}+\bm{f}(t)+\bm{a}(t)\;\;\;,$ (34)

where

$\bm{F}=-\int_{0}^{t}\gamma(t-\tau)\bm{u}(\tau)d\tau+\frac{q}{m}(\bm{u}\times\bm{B})\;\;\;.$
(35)

$\bm{B}=(0,0,B_{z})$, in the above equation is the applied magnetic field.
Thus, the $z$-direction does not experience the magnetic force. Therefore, we
consider that the motion of the Brownian particle is confined in $x-y$ plane.
The related equations of motion in terms of components of velocity are:

$\dot{u_{x}}=-\int_{0}^{t}\gamma(t-\tau)u_{x}(\tau)d\tau+\Omega
u_{y}+f_{x}(t)+a_{x}(t)$ (36)

and

$\dot{u_{y}}=-\int_{0}^{t}\gamma(t-\tau)u_{y}(\tau)d\tau-\Omega
u_{x}+f_{y}(t)+a_{y}(t)\;\;\;$ (37)

where we have used $\Omega=\frac{qB_{z}}{m}$. $a_{x}(t)$ and $a_{y}(t)$ are
the relevant components of the time-dependent force, $\bm{a}(t)$.

Now the velocity distribution function for the Langevin Eqs.(36-37) can be
written as

$P(u_{x},u_{y},u_{x}(0),u_{y}(0);t)=\left(\frac{1}{2\pi}\right)\left[\frac{1}{|\bm{\mathcal{A}(t)}|}\right]^{\frac{1}{2}}\exp\left[-\frac{1}{2}g{\dagger}(t)\bm{\mathcal{A}}^{-1}(t)g(t)\right]$
(38)

with

$g(t)=\begin{bmatrix}g_{1}(t)\\\ g_{2}(t)\end{bmatrix}\;\;.$ (39)

where

$\displaystyle g_{1}(t)$ $\displaystyle=$ $\displaystyle
u_{x}(t)-\left[\left(\chi_{1}(t)u_{x}(0)+\chi_{2}(t)u_{y}(0)\right)+\left(\int_{0}^{t}\chi_{1}(\tau)a_{x}(t-\tau)d\tau+\int_{0}^{t}\chi_{2}(\tau)a_{y}(t-\tau)d\tau\right)\right]$
(40) $\displaystyle=$ $\displaystyle u_{x}(t)-\left[c_{1}+p_{x}\right]$
$\displaystyle=$ $\displaystyle u_{x}(t)-{c_{1}}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\chi_{1}(\tau)f_{x}(t-\tau)d\tau+\int_{0}^{t}\chi_{2}(\tau)f_{y}(t-\tau)d\tau$

and

$\displaystyle g_{2}(t)$ $\displaystyle=$ $\displaystyle
u_{y}(t)-\left[\left(\chi_{1}(t)u_{y}(0)-\chi_{2}(t)u_{x}(0)\right)+\left(\int_{0}^{t}\chi_{1}(\tau)a_{y}(t-\tau)d\tau-\int_{0}^{t}\chi_{2}(\tau)a_{x}(t-\tau)d\tau\right)\right]$
(41) $\displaystyle=$ $\displaystyle u_{y}(t)-\left[c_{2}+p_{y}\right]$
$\displaystyle=$ $\displaystyle u_{x}(t)-{c_{2}}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\chi_{1}(\tau)f_{y}(t-\tau)d\tau-\int_{0}^{t}\chi_{2}(\tau)f_{x}(t-\tau)d\tau\;\;\;.$

Here we have used $c_{1}=\chi_{1}u_{x}(0)+\chi_{2}u_{y}(0)$ ,
$c_{2}=\chi_{1}u_{y}(0)-\chi_{2}u_{x}(0)$,
$p_{x}=\int_{0}^{t}\chi_{1}(\tau)a_{x}(t-\tau)d\tau+\int_{0}^{t}\chi_{2}(\tau)a_{y}(t-\tau)d\tau$
and
$p_{y}=\int_{0}^{t}\chi_{1}(\tau)a_{y}(t-\tau)d\tau-\int_{0}^{t}\chi_{2}(\tau)a_{x}(t-\tau)d\tau$.
The response functions, $\chi_{1}(t)$ and $\chi_{2}(t)$, which appear in the
above equations are inverse Laplace transform of the following relations

$\tilde{\chi}_{1}(s)=\frac{\left[s+\tilde{\gamma}(s)\right]}{\left[s+\tilde{\gamma}(s)\right]^{2}+\Omega^{2}},$
(42)

and

$\tilde{\chi}_{2}(s)=\frac{\Omega}{\left[s+\tilde{\gamma}(s)\right]^{2}+\Omega^{2}},$
(43)

respectively. Then we define the matrix, $\bm{\mathcal{A}(t)}$ with
$A_{ij}=\langle g_{i}(t)g_{j}(t)\rangle$.

We are now in a position to propose the Fokker-Planck equation for the non-
Markovian dynamics with Eqs.(36-37)21 as

$\displaystyle\frac{\partial{P}}{\partial{t}}$ $\displaystyle=$
$\displaystyle\beta_{1}\left[\nabla_{{\bf{u}}}.{\bf{u}}P\right]+\beta_{2}\left[{\bf{u}}\times\nabla_{{\bf{u}}}P\right]_{z}+G_{1}(t)\frac{\partial
P}{\partial u_{x}}+G_{2}(t)\frac{\partial P}{\partial
u_{y}}+H(t){\nabla_{{\bf{u}}}}^{2}P\;\;\;,$ (44)

where $\beta_{1}(t)$, $\beta_{2}(t)$, $H(t)$ ,$G_{1}(t)$ and $G_{2}(t)$ are
the relevant time-dependent quantities.

Following Sec. IIA, we determine these coefficients as

$\beta_{1}(t)=-\frac{(\chi_{1}\dot{\chi}_{1}+\chi_{2}\dot{\chi}_{2})}{(\chi_{1}^{2}+\chi_{2}^{2})}\;\;\;,$
(45)
$\beta_{2}(t)=-\frac{(\chi_{2}\dot{\chi_{1}}-\chi_{1}\dot{\chi}_{2})}{(\chi_{1}^{2}+\chi_{2}^{2})}\;\;\;,$
(46)
$H(t)=\frac{(\chi_{1}^{2}+\chi_{2}^{2})}{2}\frac{d}{dt}\left[\frac{A(t)}{(\chi_{1}^{2}+\chi_{2}^{2})}\right]\;\;,$
(47)

$G_{1}(t)=-\dot{p_{x}}-\beta_{1}(t)p_{x}+\beta_{2}(t)p_{y}\;\;\;,$ (48)

and

$G_{2}(t)=-\dot{p_{y}}-\beta_{1}(t)p_{y}-\beta_{2}(t)p_{x}\;\;\;.$ (49)

Making use of Eqs.(45-49) in Eq.(44), we have

$\displaystyle\frac{\partial{P}}{\partial{t}}$ $\displaystyle=$
$\displaystyle\beta_{1}(t)\nabla_{{\bf{u}}}.{\bf{u}}P-\left[\bf{\dot{p}}+\beta_{1}(t){\bf{p}}\right].\nabla_{{\bf{u}}}P-\beta_{2}(t)\left[{\bf{p}}\times\nabla_{{\bf{u}}}P\right]_{z}$
(50) $\displaystyle+$
$\displaystyle\beta_{2}(t)\left[{\bf{u}}\times\nabla_{{\bf{u}}}P\right]_{z}+H(t){\nabla_{{\bf{u}}}}^{2}P\;\;\;.$

where ${\bf{p}}=\left(p_{x},p_{y},0\right)$. This is the required Fokker-
Planck equation. Now one can check that the distribution function (38) is a
solution to the above equation. It signifies an essential appraisal of the
present calculation. Another check is that the above equation exactly
corresponds to the one which was derived in Ref.20 using the characteristic
function. Now comparing this subsection with Sec.2 in Ref.20 one may be sure
about the simplicity and the shortcuts of the present method. The plainness
may be noticeable in the additional harmonic force field, which we will
demonstrate in the next subsection.

For further check, one can show easily that in the absence of time-dependent
force, $\bm{a(t)}=0$ and the above equation reduces to the following known
result20, 21,

$\begin{array}[]{rcl}\dfrac{\partial{P}}{\partial{t}}&=&\beta_{1}(t)\left[\nabla_{{\bf{u}}}.{\bf{u}}P\right]\end{array}+\beta_{2}(t)\left[{\bf{u}}\times\nabla_{{\bf{u}}}P\right]_{z}+H(t){\nabla_{{\bf{u}}}}^{2}P\;\;\;,$
(51)

since $p_{x}=p_{y}=0$ as well as $G_{1}=G_{2}=0$.

Finally, at the Markovian limit, the Fokker-Planck equation (50) reduces to

$\dfrac{\partial{P}}{\partial{t}}=\gamma_{0}\left[\nabla_{{\bf{u}}}.{\bf{u}}P\right]-{\bf{a}.}\nabla_{{\bf{u}}}P+\Omega\left[{\bf{u}}\times\nabla_{{\bf{u}}}P\right]_{z}+\gamma_{0}k_{B}T{\nabla_{{\bf{u}}}}^{2}P$
(52)

In the absence of a time-dependent force field, $\bm{a}(t)=0$, the above
equation is the same as that obtained for the Markovian Brownian motion is
described by a charged particle across a magnetic field 45, 46. Thus the
Fokker-Planck equation (50) satisfies all possible limiting situations.
However, it is to be noted here that comparing the above equation with that of
Eqs.(50-51), one may find how the cyclotron frequency, as well as the effect
of the time-dependent external force field, is modulated in the presence of a
non-Markovian thermal bath. The modulations are quite similar to the case, the
external time-dependent force driven harmonic oscillator. Here the cyclotron
frequency plays a similar role as that of the frequency of the harmonic
oscillator. In the following subsection, we will investigate the relevant
modulations in the presence of both conservative and non-conservative force
fields, respectively.

###  II.4 Non-Markovian dynamics of a Brownian oscillator in the presence of a
magnetic field and a time-dependent external force

In the presence of a two-dimensional harmonic potential energy field, the
Langevin equations of motion (36-37) become 22

$\dot{u}_{x}=-\omega^{2}x-\int_{0}^{t}\gamma(t-\tau)u_{x}(\tau)d\tau+\Omega
u_{y}+f_{x}(t)+a_{x}(t)$ (53)

and

$\dot{u}_{y}=-\omega^{2}y-\int_{0}^{t}\gamma(t-\tau)u_{y}(\tau)d\tau-\Omega
u_{x}+f_{y}(t)+a_{y}(t)\;\;\;.$ (54)

where $\omega$ is the frequency of the harmonic oscillator. Since the Langevin
equations (53-54) of motion correspond to the Gaussian noise-driven linear
system, then the phase space distribution function is a Gaussian one 2. Using
the matrix, $\bm{\mathcal{A}}^{\prime}$ (which is defined in Appendix A) and
its inverse, the phase space distribution function can be written as:

$P(x,x(0);y,y(0);u_{x},u_{x}(0);u_{y},u_{y}(0);t)=(2\pi)^{-2}(A_{1}A_{2}-A_{3}^{2}-A_{4}^{2})^{-1}\exp\left[-\frac{1}{2}g{\dagger}(t){\bm{\mathcal{A}}^{\prime}}^{-1}(t)g(t)\right]$
(55)

with

$g(t)=\begin{bmatrix}g_{1}(t)\\\ g_{2}(t)\\\ g_{3}(t)\\\
g_{4}(t)\end{bmatrix}.$ (56)

$g_{1}(t)$, $g_{2}(t)$, $g_{3}(t)$ and $g_{4}(t)$ are defined in Appendix A.
Then, considering the previous subsections, one may propose the following
Fokker-Planck equation,

$\displaystyle\frac{\partial P}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\partial u_{x}P}{\partial x}-\frac{\partial
u_{y}P}{\partial y}-G_{1}(t)\frac{\partial P}{\partial
u_{x}}-G_{2}(t)\frac{\partial P}{\partial u_{y}}+H_{1}(t)\left[x\frac{\partial
P}{\partial u_{x}}+y\frac{\partial P}{\partial u_{y}}\right]$ $\displaystyle+$
$\displaystyle H_{2}(t)\left[\frac{\partial u_{x}P}{\partial
u_{x}}+\frac{\partial u_{y}P}{\partial
u_{y}}\right]-H_{3}(t)\left[\frac{\partial u_{y}P}{\partial
u_{x}}-\frac{\partial u_{x}P}{\partial
u_{y}}\right]-H_{4}(t)\left[x\frac{\partial P}{\partial u_{y}}-y\frac{\partial
P}{\partial u_{x}}\right]$ $\displaystyle+$ $\displaystyle
H_{5}(t)\left[\frac{\partial}{\partial x}\frac{\partial P}{\partial
u_{y}}-\frac{\partial}{\partial y}\frac{\partial P}{\partial
u_{x}}\right]+H_{6}(t)\left[\frac{\partial}{\partial x}\frac{\partial
P}{\partial u_{x}}+\frac{\partial}{\partial y}\frac{\partial P}{\partial
u_{y}}\right]+H_{7}(t)\left[\frac{\partial^{2}P}{\partial
u_{x}^{2}}+\frac{\partial^{2}P}{\partial u_{y}^{2}}\right]$

where
$G_{1}(t),G_{2}(t),H_{1}(t),H_{2}(t),H_{3}(t),H_{4}(t),H_{5}(t),H_{6}(t)$ and
$H_{7}(t)$ are relevant time-dependent quantities to account for the non-
Markovian dynamics properly. The terms with $G_{1}(t)$ and $G_{2}(t)$ appear
to consider the modulation of the effect of the time dependent force by the
harmonic, the magnetic and the dissipative force fields in the presence of the
memory effect induced feedback. Similarly, $H_{1}(t)$ ( $H_{3}(t)$) reperesnts
the modulation of the force constant of the Harmonic oscillator (the cyclotron
frequency) by the magnetic field (harmonic force field) and the frictional
memory kernel. Then we would mention that the term with $H_{2}(t)$ appears to
consider the effect of the dissipative force in the presence of magnetic field
and harmonic force. The remaining drift term with $H_{4}(t)$ is instructive
from the last subsection by virtue of the memory induced time dependent
cyclotron frequecny. Thus the Brownian particle experiences an effective time
dependent magnetic field from a time independent one in the presence of a non-
Markovian thermal bath. Then it is expected to consider a drift term with
$H_{4}(t)$ (due to an accociated induced electric field) in the Fokker-Planck
equation in phase space. At the same time, this term is instructive to
consider a diffusion term with $H_{5}(t)$ which contains cross derivatives.
This instruction is corroborated by the relevant matrix,
$\bm{\mathcal{A}}^{\prime}$. Thus the non-Markovian dynamics introduces a
special kind of cross-correlation between position and velocity in the
presence of magnetic field and conservative force. Then it is apparent that
the induced electric field may be the underlying reason for the special cross-
correlation. Finally, $H_{6}(t)$ and $H_{7}(t)$ are the usual diffusion terms
for the non-Markovian dynamics in the phase space as suggested by
$\bm{\mathcal{A}}^{\prime}$. To avoid any confusion, we would mention that the
diffusion terms with other possible cross derivatives are not considered since
the cross-correlation of the fluctuations is zero for the individual case.
However, following the previous subsections, we obtain the required Fokker-
Planck equation as

$\displaystyle\frac{\partial P}{\partial t}$ $\displaystyle=$
$\displaystyle-{\bf{u}}.\nabla_{{\bf{x}}}P-\left[\left[{\bf{\dot{p}}}+H_{2}(t){\bf{p}}+H_{1}(t){\bf{q}}\right].{\nabla_{{\bf{u}}}}P+H_{3}(t)\left[{\bf{p}}\times\nabla_{{\bf{u}}}P\right]_{z}-H_{4}(t)\left[{\bf{q}}\times\nabla_{{\bf{u}}}P\right]_{z}\right]$
(58) $\displaystyle+$ $\displaystyle
H_{1}(t){\bf{x}}.\nabla_{{\bf{u}}}P+H_{2}(t)\nabla_{{\bf{u}}}.{\bf{u}}P+H_{3}(t)\left[{\bf{u}}\times\nabla_{{\bf{u}}}P\right]_{z}-H_{4}(t)\left[{\bf{x}}\times\nabla_{{\bf{u}}}P\right]_{z}$
$\displaystyle+$ $\displaystyle
H_{5}(t)\left[\nabla_{{\bf{x}}}\times\nabla_{{\bf{u}}}P\right]_{z}+H_{6}(t)\nabla_{{\bf{u}}}.\nabla_{{\bf{x}}}P+H_{7}(t){\nabla_{{\bf{u}}}}^{2}P\;\;.$

The time-dependent coefficients in the above equations are defined in Appendix
B. Using the definition of the coefficients, one may check that the
distribution function (55) is a solution of the above Fokker-Planck equation.

For further checking of the present calculation, we consider the condition
with $\bm{a}(t)=0$. Then one can easily show that
$\bf{\dot{p}}=\bf{p}=\bf{q}=0$ and the above Fokker-Planck equation reduces to
the following known result23

$\displaystyle\frac{\partial P}{\partial t}$ $\displaystyle=$
$\displaystyle-{\bf{u}}.\nabla_{{\bf{x}}}P+H_{1}(t){\bf{x}}.\nabla_{{\bf{u}}}P+H_{2}(t)\nabla_{{\bf{u}}}.{\bf{u}}P+H_{3}(t)\left[{\bf{u}}\times\nabla_{{\bf{u}}}P\right]_{z}$
(59) $\displaystyle-$ $\displaystyle
H_{4}(t)\left[{\bf{x}}\times\nabla_{{\bf{u}}}P\right]_{z}+H_{5}(t)\left[\nabla_{{\bf{x}}}\times\nabla_{{\bf{u}}}P\right]_{z}+H_{6}(t)\nabla_{{\bf{u}}}.\nabla_{{\bf{x}}}P+H_{7}(t){\nabla_{{\bf{u}}}}^{2}P\;\;\;.$

At the same time, Eq.(58) also reduces to all the other known results at the
specific limits. Thus the accuracy of the calculation is confirmed.

Before leaving this subsection, we would mention the following points. The
Fokker-Planck equation corresponding to the equations of motion (53-54) was
derived in Ref.22 using the relevant characteristic function. This equation
was not a correct one as reported in 23. Then its revised version (Eq.(7) in
24) was published very recently. There is a typo in Eq.(7), as mentioned in
the introduction. However, some comments in 24 regarding the present method
seem to be misleading. Then we put the following discussion.

(i) To apply the present method, one may not need prior knowledge based on the
characteristic function. For example, the proposed Fokker-Planck equations in
Ref.23, 53 contain extra terms compared to the required Fokker-Planck
equation. Applying two simple mathematical notions [(1) Setting of linear
algebraic equations with the collection of coefficients of phase space
variables and their appropriate multiples after putting the given distribution
function into the proposed Fokker-Planck equation, and (2) solution of the
algebraic equations by the elimination procedure.], one may anchor at the
required FPE. It leads to remove the inconsistencies with two drift terms and
a diffusion term in the Fokker-Planck equation (61) in Ref.22. Thus the method
seems to be a straightforward as well as shortcut one compared to the other
solution-based methods18, 19. It may be instructive to derive the Fokker-
Planck equation in a shortcut way for complex systems like the magnetic field
in an arbitrary direction.

(ii) The present method may be helpful for a more complex case like a
multidimensional system with relevant position-dependent couplings. In that
case, the drift term from the harmonic force field has to be split. Thus the
present method may be pertinent for all possible Gaussian noise-driven linear
systems for which the distribution function is known. The relevant matrix
(with the second moments corresponding to the component of fluctuations) in
the distribution function may be diagonal or not. For example, in subsections
II A and II C, the matrix is diagonal, and all the off-diagonal elements may
not be zero in subsections II B and D, respectively.

(iii) Then we address the issue regarding the shorcut way of the method.
Comparing between Sec.IIC of the present manuscript and Sec.2 plus alied
Appendix B in 20, one may be sure about the shorcutness of the present
calculation. To avoid any confusion, we note that even some algebra was not
explicit in this reference. Specifically, the transformation of Eq.(B.10) into
Eq.(B.22) with the use of Eqs.(B.9) and (B.17-B.18) requires a lengthy
algebra. It is to be noted here that in the Sec.IIC, three algebraic relations
among five unknown, $\beta_{1}(t),\beta_{2}(t),G_{1}(t),G_{2}(t)$ and $H(t)$
are implicit. One may obtain these relations upon substitution of the
distribution function(38) in Eq.(44). $\beta_{1}(t),\beta_{2}(t)$ and $H(t)$
are independent of the external time-dependent force field. Using this
property, one can easily determine these quantities as reported in 21. Then
determination of $G_{1}(t)$ and $G_{2}(t)$ is very simple. Thus the relevant
algebra is almost the same with or without the external time-dependent force
field. Thus the present method is too shortcut (in adition to its simplicity)
compared to other method. Furthermore, one may compare Ref.40 with Sec.III
allied with appendix A in 22 to avoid any confusion regarding the present
context. Again we note that a very long and careful algebra was implicit, as
mentioned in the reference 22. It is to be noted here that the algebraic
equations in the appendix in Ref.40 were solved by a simple inspection. This
leads to determine $G_{1}(t)$ and $G_{2}(t)$ in Eq.(LABEL:eq84) in Sec.IID
with a bit of additional effort to derive the relations among the coefficients
in this equation. Thus there should be no doubt about the shortcut way of the
present method.

###  II.5 Memory induced electric field from time-independent magnetic field

There is no particular emphasis in all the relevant references 21, 22, 23, 24
to interpret the physical significance of the additional terms (in the
required Fokker-Planck equations) due to the applied magnetic field. This
subsection includes an approach to this issue.

In the presence of a time-dependent magnetic field ($\bm{B_{t}}=(0,0,B_{t})$),
the Langevin equations of motion (53-54) can be written as 33

$\dot{u}_{x}=-\omega^{2}x-\int_{0}^{t}\gamma(t-\tau)u_{x}(\tau)d\tau+\Omega_{t}u_{y}+\frac{\dot{B_{t}}y}{2}+f_{x}(t)+a_{x}(t)$
(60)

and

$\dot{u}_{y}=-\omega^{2}y-\int_{0}^{t}\gamma(t-\tau)u_{y}(\tau)d\tau-\Omega_{t}u_{x}-\frac{\dot{B_{t}}y}{2}+f_{y}(t)+a_{y}(t)\;\;,$
(61)

where $\Omega_{t}=\frac{qB_{t}}{m}$. At the Markovian limit, the overhead
equations of motion evolves as:

$\dot{u}_{x}=-\omega^{2}x-\gamma_{0}u_{x}+\Omega_{t}u_{y}+\frac{\dot{B_{t}}y}{2}+f_{x}(t)+a_{x}(t)$
(62)

and

$\dot{u}_{y}=-\omega^{2}y-\gamma_{0}u_{y}-\Omega_{t}u_{x}-\frac{\dot{B_{t}}y}{2}+f_{y}(t)+a_{y}(t)\;\;.$
(63)

Now making use of the respective Taylor series into the both sides of the
Smoluchowski integral equaion (which is the Chapman-Kolmogorov equation for a
Markov process 32), essentially due to Einstein 54, corresponding to the above
equations of motion, we obtain the following Fokker-Planck equation,

$\displaystyle\frac{\partial P}{\partial t}$ $\displaystyle=$
$\displaystyle-{\bf{u}}.\nabla_{{\bf{x}}}P+\gamma_{0}\nabla_{{\bf{u}}}.{\bf{u}}P-\bf{a}.\nabla_{{\bf{u}}}P+\omega^{2}{\bf{x}}.\nabla_{{\bf{u}}}P+\Omega_{t}\left[{\bf{u}}\times\nabla_{{\bf{u}}}P\right]_{z}$
(64) $\displaystyle+$
$\displaystyle\frac{\dot{B_{t}}}{2}\left[{\bf{x}}\times\nabla_{{\bf{u}}}P\right]_{z}+\gamma_{0}k_{B}T{\nabla_{{\bf{u}}}}^{2}P\;\;.$

Adding the noise term to the relevant Liouville equation and then doing proper
averaging of the equation55, one may easily arive at this equation. However,
in the absence of the external time-dependent force field, the above equation
evolves to

$\displaystyle\frac{\partial P}{\partial t}$ $\displaystyle=$
$\displaystyle-{\bf{u}}.\nabla_{{\bf{x}}}P+\gamma_{0}\nabla_{{\bf{u}}}.{\bf{u}}P+\omega^{2}{\bf{x}}.\nabla_{{\bf{u}}}P+\Omega_{t}\left[{\bf{u}}\times\nabla_{{\bf{u}}}P\right]_{z}$
(65) $\displaystyle+$
$\displaystyle\frac{\dot{B_{t}}}{2}\left[{\bf{x}}\times\nabla_{{\bf{u}}}P\right]_{z}+\gamma_{0}k_{B}T{\nabla_{{\bf{u}}}}^{2}P\;\;.$

Similarly, the Fokker-Planck equation at the Markovian limit for the Langevin
equations (53-54) of motion can be written as

$\displaystyle\frac{\partial P}{\partial t}$ $\displaystyle=$
$\displaystyle-{\bf{u}}.\nabla_{{\bf{x}}}P-\bf{a}.\nabla_{{\bf{u}}}P+\omega^{2}{\bf{x}}.\nabla_{{\bf{u}}}P+\gamma_{0}\nabla_{{\bf{u}}}.{\bf{u}}P+\Omega\left[{\bf{u}}\times\nabla_{{\bf{u}}}P\right]_{z}+\gamma_{0}k_{B}T{\nabla_{{\bf{u}}}}^{2}P\;\;.$
(66)

Now one may discourse on the following points in order. First, comparing
Eq.(59) with Eq.(65), we find that the coefficient of
$\left[{\bf{u}}\times\nabla_{{\bf{u}}}P\right]_{z}$ is time-dependent in both
equations. Then it is apparent in Eq.(59) that the time-independent magnetic
field behaves effectively as a time-dependent one in the presence of a non-
Markovian thermal bath.

Then one may expect the frictional memory kernel induced electric field from
the time-independent magnetic field in the phase space description. In this
occurrence, we would look at Eq.(65). The term,
$\frac{\dot{B_{t}}}{2}\left[{\bf{x}}\times\nabla_{{\bf{u}}}P\right]_{z}$ in
the above equation is due to the induced electric field from the time-
dependent magnetic field. Then one may identify the term, with the coefficient
$H_{4}\left[{\bf{x}}\times\nabla_{{\bf{u}}}P\right]_{z}$ in Eq. (59) as the
memory-induced electric field from the time-independent magnetic field. Here
the conservative force field from the harmonic potential has an important role
since, in the absence, the induced electric field may not emerge as justified
in Eq.(50). Second, analogizing among the Fokker-Planck Eqs.(50, 58,64), one
may interpret that the drift term,
$H_{4}(t)\left[{\bf{q}}\times\nabla_{{\bf{u}}}P\right]_{z}$ in Eq.(58) is due
to the modulation of the effect of the time-dependent external force field by
the memory induced electric field. Thus, the second term in the right hand
side exhibits how the exffect can be modulated by frictional memory kernel,
conservative force, constant magnetic field and induced electric field.
Finally, the induced electric field may imply to appear the cross
correlations, $<xu_{y}>$ and $<yu_{x}>$, respectively. The term,
$H_{5}(t)\left[\nabla_{{\bf{x}}}\times\nabla_{{\bf{u}}}P\right]_{z}$ in
Eqs.(58-59) is a signature of these kind of special cross correlations due to
the memory effect in the presence of magnetic field induced velocity-dependent
coupling and harmonic force as suggested by the matrix element, $A_{4}$ in
matrix, $\bm{\mathcal{A}}^{\prime}$ in Eq.(80). These are important benefits
of the derivation of the Fokker-Planck equation. The induced electric field
and related effects may be overlooked from the Langevin equations of motion.
Even the existence of these quantities may be identified with the help of the
distribution function. Still, the lengthy expression of the PFE equation
implies that the identification can not determine the effects’ contribution to
the probability flux quantitatively. According to the famous assertion of van
Kampen 9, the identification of the induced electric field demands to re-
investigate the barrier crossing dynamics 7, 48, 49, 50, 33, 51 and others 5,
52, where it was considered that a charged Brownian particle is coupled to a
Markovian thermal bath in the presence of magnetic field and conservative
force.

##  III Conclusion

In the presnt study we demonstrate that how the solution method, proposed by
das et. al.21, 23 can be used to determine the modulation of the drift term
due to an external deterministic force field in the presence of a frictional
meory kernel induced velocity dependent feedback. It is to be noted here that
the additional drift term (which is due to the external time-dependent force)
in the proposed Fokker-Planck equation creates a difficulty in using the
solution method 21. The number of independent relations (a set of linear
algebraic equations) among the coefficients which appear in the proposed
equation is less than that of the number of relevant unknowns. In this
circumstance, we need additional conditions based on the physics of the given
system. The solution of the Langevin equation implies that the response
function or the susceptibility does not depend on the external force field.
Then we consider that the drift terms for the other force fields and the
relevant diffusion terms are independent of the external force field. Using
this property into the independent relations we determine the modulation of
the drift term due to an external deterministic force field in the presence of
a frictional memory kernel induced velocity dependent feedback. We show that
this technique works even in the presence of both conservative and non-
conservative fields, respectively. With four examples, we show that the method
works well in this context. This calculation is instructive to determine
easily a diffusion term due to an external stochastic force driven Brownian
particle in the presence of frictional memory induced velocity dependent
feedback, harmonic force and magnetic field. Finally, one may find a
chronological development of the relevant Fokker-Planck equations such that it
will be helpful to understand them.

Here it is to be noted that at the present state of knowledge, some of the
terms in the FPE for the non-Markovian dynamics in the presence of a magnetic
field22, 24, 23 require special attention. In this context, an essential
identification is that the non-Markovian dynamics may induce an electric field
from the time-independent magnetic field in the presence of a conservative
force field. It may be the reason for appearing the special cross-correlation
between the position and the velocity. Thus induced electric field introduces
an additional diffusion term with a cross-derivative. At the same time, it may
modulate other diffusion terms and the effect from a time-dependent external
force field.

Before we terminate, the following points merit due concentration. First, one
may apply the present method to consider where the magnetic field may be in an
arbitrary direction. It seems to be an open problem. In this context, Sec. IID
may be instructive in deriving the Fokker-Planck equation with the least
effort. Second, the magnetic field induced additional drift and diffusion
terms in the company of a non-Markovian thermal bath may be pretty interesting
in many contexts. Then the consideration of the time-dependent deterministic
force field makes the present study very relevant in the field of stochastic
thermodynamics, which is now at an early stage of considering the non-
Markovian dynamics38, 27. The definition of work in ST requires a time-
dependent deterministic force field to drive the system from a given
equilibrium state38, 27, 39. Then the present study may be applicable to
investigate many aspects of stochastic thermodynamics, such as (i) how the
time asymmetry inherent to irreversible processes depends on the non-Markovian
dynamics? (ii) the effect of non-Markovian dynamics on the relative entropy
and dispersed works, (iii) work produced from a single reservoir, (iv)
characterization of a non-equilibrium steady state in the presence of a non-
Markovian thermal bath etc.

Third, in recent technology, the study of the ion-conducting electrolytic
materials is a significant area in physics and chemistry56, 57. The materials
have potential applications in a diverse range of all-solid-state devices,
such as rechargeable lithium batteries, flexible electrochromic displays and
smart windows56. The properties of the electrolytes are tuned by varying
chemical composition to a large extent and hence are adapted to specific needs
57. High ionic conductivity is needed for optimizing the glassy electrolytes
in various applications. Then it would be fascinating to tune the ionic
conductivity according to a specific need by a physical method. In this
context, very recent studies 31, 48, 49, 50, 33 shows that an applied magnetic
field can tune the conductivity of electrolytic material. To adjust the
conductivity of ions in the solid electrolytes, the combination of both
magnetic field and time-dependent electric field may be an important choice.
Thus the present study may find crucial applications in studying the barrier
crossing dynamics 31, 48.

Finally, based on the recent developments on the noise-induced transition in a
fluctuating magnetic field-driven harmonic oscillator, one may anticipate that
the memory-induced electric field may render the transition even in an
additive noise-driven linear system. The relevant issues are in progress.

##  IV Acknowledgment

J. Das is thankful to the University Grants Commission, Government of India
for partial financial support. M. Biswas is happy to acknowledge JRF from the
Council of Scientific and Industrial Research, Government of India. D. Mondal
thanks SERB (Project No. ECR/2018/002830/CS), Department of Science and
Technology, Government of India, for financial support and IIT Tirupati for
the new faculty seed grant.

##  V DATA AVAILABILITY

The data that support the findings of this study are available within the
article.

##  Appendix A Solution of Eqs. (53-54) and related quantities

Using the Laplace transformation the solution of Eqs. (53-54) can be written
as

$\displaystyle g_{1}(t)$ $\displaystyle=$ $\displaystyle
x(t)-\left(A(t)x(0)-B(t)y(0)+C(t)u_{x}(0)+D(t)u_{y}(0)\right)$ (67)
$\displaystyle-$
$\displaystyle\left(\int_{0}^{t}H_{0}(t-\tau)a_{x}(\tau)d\tau-\Omega^{2}\int_{0}^{t}H_{0}^{\prime}(t-\tau)a_{x}(\tau)d\tau+\Omega\int_{0}^{t}H(t-\tau)a_{y}(\tau)d\tau\right)$
$\displaystyle=$ $\displaystyle x(t)-\left[c_{1}+q_{x}\right]$
$\displaystyle=$ $\displaystyle x(t)-{c_{1}}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}H_{0}(t-\tau)f_{x}(\tau)d\tau-\Omega^{2}\int_{0}^{t}H_{0}^{\prime}(t-\tau)f_{x}(\tau)d\tau+\Omega\int_{0}^{t}H(t-\tau)f_{y}(\tau)d\tau\;\;\;,$

$\displaystyle g_{2}(t)$ $\displaystyle=$ $\displaystyle
y(t)-\left(A(t)y(0)+B(t)x(0)+C(t)u_{y}(0)-D(t)u_{x}(0)\right)$ (68)
$\displaystyle-$
$\displaystyle\left(\int_{0}^{t}H_{0}(t-\tau)a_{y}(\tau)d\tau-\Omega^{2}\int_{0}^{t}H_{0}^{\prime}(t-\tau)a_{y}(\tau)d\tau-\Omega\int_{0}^{t}H(t-\tau)a_{x}(\tau)d\tau\right)$
$\displaystyle=$ $\displaystyle y(t)-\left[c_{2}+q_{y}\right]$
$\displaystyle=$ $\displaystyle y(t)-{c_{2}}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}H_{0}(t-\tau)f_{y}(\tau)d\tau-\Omega^{2}\int_{0}^{t}H_{0}^{\prime}(t-\tau)f_{y}(\tau)d\tau-\Omega\int_{0}^{t}H(t-\tau)f_{x}(\tau)d\tau\;\;\;,$

$\displaystyle g_{3}(t)$ $\displaystyle=$ $\displaystyle
u_{x}(t)-\left(\dot{A}(t)x(0)-\dot{B}(t)y(0)+\dot{C}(t)u_{x}(0)+\dot{D}(t)u_{y}(0)\right)$
(69) $\displaystyle-$
$\displaystyle\left(\int_{0}^{t}\dot{H_{0}}(t-\tau)a_{x}(\tau)d\tau-\Omega^{2}\int_{0}^{t}\dot{H_{0}^{\prime}}(t-\tau)a_{x}(\tau)d\tau+\Omega\int_{0}^{t}\dot{H}(t-\tau)a_{y}(\tau)d\tau\right)$
$\displaystyle=$ $\displaystyle u_{x}(t)-\left[c_{3}+p_{x}\right]$
$\displaystyle=$ $\displaystyle u_{x}(t)-{c_{3}}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\dot{H_{0}}(t-\tau)f_{x}(\tau)d\tau-\Omega^{2}\int_{0}^{t}\dot{H_{0}^{\prime}}(t-\tau)f_{x}(\tau)d\tau+\Omega\int_{0}^{t}\dot{H}(t-\tau)f_{y}(\tau)d\tau\;\;\;,$

$\displaystyle g_{4}(t)$ $\displaystyle=$ $\displaystyle
u_{y}(t)-\left(\dot{A}(t)y(0)+\dot{B}(t)x(0)+\dot{C}(t)u_{y}(0)-\dot{D}(t)u_{x}(0)\right)$
(70) $\displaystyle-$
$\displaystyle\left(\int_{0}^{t}\dot{H_{0}}(t-\tau)a_{y}(\tau)d\tau-\Omega^{2}\int_{0}^{t}\dot{H_{0}^{\prime}}(t-\tau)a_{y}(\tau)d\tau-\Omega\int_{0}^{t}\dot{H}(t-\tau)a_{x}(\tau)d\tau\right)$
$\displaystyle=$ $\displaystyle u_{y}(t)-\left[c_{4}+p_{y}\right]$
$\displaystyle=$ $\displaystyle u_{y}(t)-{c_{4}}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\dot{H_{0}}(t-\tau)f_{y}(\tau)d\tau-\Omega^{2}\int_{0}^{t}\dot{H_{0}^{\prime}}(t-\tau)f_{y}(\tau)d\tau-\Omega\int_{0}^{t}\dot{H}(t-\tau)f_{x}(\tau)d\tau\;\;\;,$

where

$A\equiv A(t)=\chi_{0}(t)+\Omega^{2}\omega^{2}\chi(t)\;\;,$ (71)

$B\equiv B(t)=\Omega\omega^{2}H^{\prime}(t)\;\;,$ (72) $C\equiv
C(t)=H_{0}(t)-\Omega^{2}H_{0}^{\prime}(t)\;\;,$ (73)

$D\equiv D(t)=\Omega H(t)\;\;,$ (74)

with

$\chi_{0}(t)=1-\omega^{2}\int_{0}^{t}H_{0}(\tau)d\tau\;\;,$ (75)

$\chi(t)=\int_{0}^{t}H_{0}^{\prime}(\tau)d\tau\;\;.$ (76)

The above Eqs.(75-76) implies that $\chi_{0}(0)=1.0$ and $\chi(0)=0$.

Now we have to define the functions $H_{0}(t)$, $H_{0}^{\prime}(t)$, $H(t)$
and $H^{\prime}(t)$ which appear in the above equations. These are the inverse
Laplace transformation of $\tilde{H_{0}}(s)$, $\tilde{H_{0}^{\prime}}(s)$,
$\tilde{H}(s)$ and $\tilde{H^{\prime}}(s)$, respectively. Here we have used
$\tilde{H}(s)=s\tilde{H^{\prime}}(s)$ and
$\tilde{H_{0}^{\prime}}(s)=s^{2}\tilde{H_{00}}(s)$. $\tilde{H_{0}}(s)$,
$\tilde{H^{\prime}}(s)$ and $\tilde{H_{00}}(s)$ are defined as

$\tilde{H}_{0}(s)=\frac{1}{s^{2}+s\tilde{\gamma}(s)+\omega^{2}}\;\;,$ (77)

$\tilde{H^{\prime}}(s)=\frac{1}{(s^{2}+s\tilde{\gamma}(s)+\omega^{2})^{2}+(\Omega
s)^{2}}\;\;,$ (78)

and

$\tilde{H_{00}}(s)=\frac{1}{(s^{2}+s\tilde{\gamma}(s)+\omega^{2})[(s^{2}+s\tilde{\gamma}(s)+\omega^{2})^{2}+(\Omega
s)^{2}]}\;\;\;.$ (79)

Here $\tilde{\gamma}(s)$ is the Laplace transform of $\gamma(t)$.

We now consider the fluctuations in position and velocity, respectively. All
the second moments corresponding to the fluctuations (as given by Eqs.(67-70))
can be represented by the matrix, $\bm{\mathcal{A}}^{\prime}(t)$ with
$A_{ij}^{\prime}=\langle g_{i}(t)g_{j}(t)\rangle$. Using Eqs.(67-70), one can
read the matrix $\bm{\mathcal{A}}^{\prime}(t)$ as

$\bm{\mathcal{A}}^{\prime}=\begin{bmatrix}A_{1}&0&A_{3}&A_{4}\\\
0&A_{1}&-A_{4}&A_{3}\\\ A_{3}&-A_{4}&A_{2}&0\\\
A_{4}&A_{3}&0&A_{2}\end{bmatrix}$ (80)

where we have used $A_{11}^{\prime}=A_{22}^{\prime}=A_{1}$,
$A_{33}^{\prime}=A_{44}^{\prime}=A_{2}$,
$A_{13}^{\prime}=A_{31}^{\prime}=A_{24}^{\prime}=A_{42}^{\prime}=A_{3}$,
$A_{14}^{\prime}=A_{41}^{\prime}=A_{4}$, and
$A_{23}^{\prime}=A_{32}^{\prime}=-A_{4}$. Then the inverse of the above matrix
can be written as

${\bm{\mathcal{A}}^{\prime}}^{-1}(t)=\frac{adj(\bm{\mathcal{A}}^{\prime}(t))}{|\bm{\mathcal{A}}^{\prime}(t)|}=\frac{1}{(A_{1}A_{2}-A_{3}^{2}-A_{4}^{2})}\begin{bmatrix}A_{2}&0&-A_{3}&-A_{4}\\\
0&A_{2}&A_{4}&-A_{3}\\\ -A_{3}&A_{4}&A_{1}&0\\\
-A_{4}&-A_{3}&0&A_{1}\end{bmatrix}$ (81)

##  Appendix B Time-dependent coefficients of the Fokker-Planck equation
(LABEL:eq84)

Following Ref.40 we use the distribution function (55) in Eq. (LABEL:eq84) and
then get the time dependent coefficients as

$H_{1}=\frac{\left[-\ddot{A}\tilde{a}_{x}(t)-\ddot{B}\tilde{a}_{y}(t)-\ddot{C}\tilde{a}_{v_{x}}(t)-\ddot{D}\tilde{a}_{v_{y}}(t))\right]}{\Delta_{m}},$
(82)
$H_{2}=\frac{\left[\ddot{A}\tilde{b}_{x}(t)-\ddot{B}\tilde{b}_{y}(t)-\ddot{C}\tilde{b}_{v_{x}}(t)-\ddot{D}\tilde{b}_{v_{y}}(t))\right]}{\Delta_{m}},$
(83)
$H_{3}=\frac{\left[-\ddot{A}\tilde{d}_{x}(t)+\ddot{B}\tilde{d}_{y}(t)-\ddot{C}\tilde{d}_{v_{x}}(t)+\ddot{D}\tilde{d}_{v_{y}}(t))\right]}{\Delta_{m}},$
(84)

$H_{4}=\frac{\left[-\ddot{A}\tilde{c}_{x}(t)+\ddot{B}\tilde{c}_{y}(t)+\ddot{C}\tilde{c}_{v_{x}}(t)-\ddot{D}\tilde{c}_{v_{y}}(t))\right]}{\Delta_{m}},$
(85) $H_{5}=\left[\dot{A_{4}}+H_{2}A_{4}+H_{3}A_{3}-H_{6}A_{1}\right],$ (86)
$H_{6}=\left[\dot{A_{3}}-A_{2}-H_{3}A_{4}+H_{1}A_{1}+H_{2}A_{3}\right],$ (87)

$H_{7}=\frac{1}{2}\left[\dot{A_{2}}+2H_{1}A_{3}+2H_{2}A_{2}-2H_{6}A_{4}\right],$
(88)

Here we have used

$\Delta_{m}=(A^{2}+B^{2})(\dot{C}^{2}+\dot{D}^{2})+(C^{2}+D^{2})(\dot{A}^{2}+\dot{B}^{2})-2(AC-
BD)(\dot{A}\dot{C}-\dot{B}\dot{D})-2(AD+BC)(\dot{A}\dot{D}+\dot{B}\dot{C})\;\;\;,$
(89)

$\tilde{a}_{x}(t)=A(\dot{C}^{2}+\dot{D}^{2})-C(\dot{A}\dot{C}-\dot{B}\dot{D})-D(\dot{A}\dot{D}+\dot{B}\dot{C}),$

$\tilde{c}_{x}(t)=B(\dot{C}^{2}+\dot{D}^{2})+D(\dot{A}\dot{C}-\dot{B}\dot{D})-C(\dot{A}\dot{D}+\dot{B}\dot{C}),$

$\tilde{b}_{x}(t)=B(C\dot{D}-\dot{C}D)+C(A\dot{C}-\dot{A}C)+D(A\dot{D}-D\dot{A}),$

$\tilde{d}_{x}(t)=B(C\dot{C}+D\dot{D})-C(A\dot{D}+C\dot{B})+D(A\dot{C}-D\dot{B}).$
(90)

$\tilde{a}_{y}(t)=B(\dot{C}^{2}+\dot{D}^{2})+D(\dot{A}\dot{C}-\dot{B}\dot{D})-C(\dot{A}\dot{D}+\dot{B}\dot{C}),$

$\tilde{c}_{y}(t)=A(\dot{C}^{2}+\dot{D}^{2})-C(\dot{A}\dot{C}-\dot{B}\dot{D})-D(\dot{A}\dot{D}+\dot{B}\dot{C}),$

$\tilde{b}_{y}(t)=A(C\dot{D}-\dot{C}D)-C(B\dot{C}-\dot{B}C)-D(B\dot{D}-\dot{B}D),$

$\tilde{d}_{y}(t)=A(C\dot{C}+\dot{D}D)+C(B\dot{D}-\dot{A}C)-D(B\dot{C}+\dot{A}D).$
(91)

$\tilde{a}_{v_{x}}(t)=C(\dot{A}^{2}+\dot{B}^{2})-A(\dot{A}\dot{C}-\dot{B}\dot{D})-B(\dot{A}\dot{D}+\dot{B}\dot{C}),$

$\tilde{c}_{v_{x}}(t)=D(\dot{A}^{2}+\dot{B}^{2})+B(\dot{A}\dot{C}-\dot{B}\dot{D})-A(\dot{A}\dot{D}+\dot{B}\dot{C}),$

$\tilde{b}_{v_{x}}(t)=A(A\dot{C}-\dot{A}C)+B(B\dot{C}-\dot{B}C)-D(A\dot{B}-\dot{A}B),$

$\tilde{d}_{v_{x}}(t)=A(A\dot{D}+\dot{B}C)+B(B\dot{D}-\dot{A}C)-D(A\dot{A}+\dot{B}B).$
(92)

$\tilde{a}_{v_{y}}(t)=D(\dot{A}^{2}+\dot{B}^{2})+B(\dot{A}\dot{C}-\dot{B}\dot{D})-A(\dot{A}\dot{D}+\dot{B}\dot{C}),$

$\tilde{c}_{v_{y}}(t)=C(\dot{A}^{2}+\dot{B}^{2})-A(\dot{A}\dot{C}-\dot{B}\dot{D})-B(\dot{A}\dot{D}+\dot{B}\dot{C}),$

$\tilde{b}_{v_{y}}(t)=A(A\dot{D}-\dot{A}D)+B(B\dot{D}-\dot{B}D)+C(A\dot{B}-\dot{A}B),$

$\tilde{d}_{v_{y}}(t)=A(A\dot{C}-\dot{B}D)+B(B\dot{C}+\dot{A}D)-C(A\dot{A}+\dot{B}B).$
(93)

It is to be noted here that all the time-dependent coefficients (as given by
Eqs.(82-88)) do not depend on $c_{1}$, $c_{2}$, $c_{3}$and $c_{4}$ as like as
the previous cases with $\bm{a}(t)=0$. Then following the earlier cases we
have

$G_{1}=\dot{p_{x}}+H_{1}q_{x}+H_{2}p_{x}-H_{3}p_{y}+H_{4}q_{y},$ (94)

$G_{2}=\dot{p_{y}}+H_{1}q_{y}+H_{2}p_{y}+H_{3}p_{x}-H_{4}q_{x},$ (95)

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