Non-markovian dynamics-induced fictitious electric field in the presence of a time-independent magnetic field

Abstract

Using the property of the solution of the Langevin dynamics with a generalized frictional memory kernel and time-dependent deterministic force field, we show that a solution method (which is very simple as well as shortcut) can be used to derive a Fokker–Planck-like equation (FPLE) for this dynamics. Using this equation, we derive a relation to find the modulation of the entropy production by a time-dependent external force. We also derive a relation between the phase space dynamics and the entropy production of the irreversible thermodynamics. Then FPLE 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 FPLE 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.Another very important point is to be noted here. To interpret the FPLE, we recognize that the memory of the non-Markovian dynamics can induce a fictitious electric field in the presence of a time-independent magnetic field and a conservative force field. Then one may notice that like the forces from time-dependent damping, harmonic potential and magnetic field, the fictitious electric field puts its own identity to modulate the effect of the external time-dependent force in the presence of a non-Markovian thermal bath. We understand that the present study, with the recognition of the induced electric field-like quantity, will bring strong attention to different areas of non-equilibrium statistical mechanics, such as the physical tuning of conductivity of ions in solid electrolytes and stochastic thermodynamics for non-Markovian dynamics, etc.

Graphical abstract

Appendices

Non-Markovian dynamics of time-dependent force-driven harmonic oscillator

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

$$\begin{aligned} \ddot{{\varvec{x}}}+\omega ^{2}{\varvec{x}}(t)+\int _{0}^{t}\gamma (t-\tau )\dot{{\varvec{x}}}(\tau ){\text {d}}\tau ={\varvec{f}}(t)+{\varvec{a}}(t), \end{aligned}$$

(A1)

where ${\varvec{x}}$ corresponds to the relevant position vector.

Now following Sect. 2.1, the probability density function for the Langevin equation of motion [Eq. (A1)] can be written as [9, 16, 40]

$$\begin{aligned}{} & {} P({\varvec{x}},{\varvec{u}},{\varvec{x}}(0),{\varvec{u}}(0);t)=\left( \frac{1}{2\pi }\right) ^{3}\left[ \frac{1}{|\varvec{{\mathcal {A}}(t)}|}\right] ^{\frac{3}{2}}\nonumber \\{} & {} \quad \times \exp \left[ -\frac{3}{2}{\varvec{g}}\dag (t)\varvec{{\mathcal {A}}(t)}^{-1}{\varvec{g}}(t)\right] \end{aligned}$$

(A2)

with

$$\begin{aligned} {\varvec{g}}(t)=\begin{bmatrix}\varvec{g_{1}}(t)\\ \varvec{g_{2}}(t) \end{bmatrix}, \end{aligned}$$

(A3)

where

$$\begin{aligned} \varvec{g_{1}}(t)= & {} {\varvec{x}}(t)-\left[ \chi _{{\varvec{x}}}(t){\varvec{x}}_{0}+\chi _{{\varvec{u}}}(t){\varvec{u}}_{0}-\int _{0}^{t}\chi _{{\varvec{u}}}(\tau ){\varvec{a}}(t-\tau ){\text {d}}\tau \right] \nonumber \\= & {} {\varvec{x}}(t)-\left[ \varvec{c_{1}}+\varvec{q_{x}}\right] =\int _{0}^{t}\chi _{{\varvec{u}}}(\tau ){\varvec{f}}(t-\tau ){\text {d}}\tau . \end{aligned}$$

(A4)

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

$$\begin{aligned} \varvec{g_{2}}(t)= & {} {\varvec{u}}(t)-\left[ \dot{\chi }_{{\varvec{x}}}(t){\varvec{x}}_{0}+\dot{\chi }_{{\varvec{u}}}(t){\varvec{u}}_{0}-\int _{0}^{t}\dot{\chi }_{{\varvec{u}}}(\tau ){\varvec{a}}(t-\tau ){\text {d}}\tau \right] \nonumber \\= & {} {\varvec{u}}(t)-[\dot{\varvec{c_{1}}}+\varvec{p_{x}}]=\int _{0}^{t}\dot{\chi }_{{\varvec{u}}}(\tau ){\varvec{f}}(t-\tau ){\text {d}}\tau . \end{aligned}$$

(A5)

Here, we have used $\varvec{c_{1}}=\left[ \chi _{{\varvec{x}}}(t){\varvec{x}}_{0}+\chi _{{\varvec{u}}}(t){\varvec{u}}_{0}\right]$, $\varvec{q_{x}}=\int _{0}^{t}\chi _{{\varvec{u}}}(\tau ){\varvec{a}}(t-\tau ){\text {d}}\tau$ and $\varvec{\dot{q_{x}}}=\varvec{p_{x}}$. $\chi _{{\varvec{x}}}(t)$ and $\chi _{{\varvec{u}}}(t)$ in Eq. (A4) are the inverse Laplace transform of the relations, ${\tilde{\chi }}_{{\varvec{x}}}(s)=\frac{{\tilde{\gamma }}(s)+s}{s^{2}+s{\tilde{\gamma }}(s)+\omega ^{2}}$ and ${\tilde{\chi }}_{{\varvec{u}}}(s)=\frac{1}{s^{2}+s{\tilde{\gamma }}(s)+\omega ^{2}}$, respectively. Then, we define the matrix $\varvec{{\mathcal {A}}(t)}$ with $A_{ij}=\langle \varvec{g_{i}}(t)\cdot \varvec{g_{j}}(t)\rangle$.

Now, the nature of the distribution function leads to propose the following FPLE as an equivalent description of Eq. (A1),

$$\begin{aligned} \frac{\partial {P}}{\partial {t}}= & {} -{\varvec{u}}\cdot \nabla _{\mathbf{{x}}}P+{\varvec{G}}(t).\nabla _{\mathbf{{u}}}P+H_{1}(t){\varvec{x}}\cdot \nabla _{\mathbf{{u}}}+H_{2}(t)\nabla _{\mathbf{{u}}}.\mathbf{{u}}P\nonumber \\{} & {} \quad + H_{3}(t)\nabla _{\mathbf{{u}}}.\nabla _{\mathbf{{x}}}P+H_{4}(t){\nabla _{\mathbf{{u}}}}^{2}P, \end{aligned}$$

(A6)

where ${\varvec{G}}(t), H_{1}(t), H_{2}(t), H_{3}(t)$ and $H_{4}(t)$ are the relevant time-dependent coefficients to account for the non-Markovian dynamics 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 cannot 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 $H_{1}(t)$. From the dynamics of a damped harmonic oscillator, we know that the frequency of the oscillator is modulated by the damping since the oscillation is opposed by the frictional force. To account for the modulation by the time-dependent damping strength, we consider the coefficient is a time-dependent one. The remaining drift term is due to the dissipative force. Its coefficent, $H_2(t)$ is assumed to be time-dependent for the time-dependent damping strength. We now consider the inclusion of the diffusion terms. It is well known [16, 38] that the non-Markovian dynamics induces a correlation between the cannonical conjugate pair. This correlation is instructive to include the diffusion term, $H_{3}(t)\nabla _{\mathbf{{u}}}.\nabla _{\mathbf{{x}}}P$. Finally, the last term is the usual diffusion term. To avoid any confusion, we would mention here that following Refs. [40, 42, 88], 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

$$\begin{aligned} {\varvec{G}}(t)=-\dot{\varvec{p_{x}}}-H_{2}(t)\varvec{p_{x}}-H_{1}(t)\varvec{q_{x}}, \end{aligned}$$

(A7)

and $H_{1}(t)={\tilde{\omega }}^{2}(t)$, $H_{2}(t)={\tilde{\beta }}(t)$, $H_{3}(t)=\frac{k_{\text {B}}T}{\omega ^2}\left[ {\tilde{\omega }}^{2}(t)-\omega ^2\right]$ $H_{4}(t)=\frac{k_{\text {B}}T}{\omega ^2}{\tilde{\beta }}(t)$. Here, we have used ${\tilde{\beta }}(t)=-\frac{d\ln \Delta (t)}{{\text {d}}t}$, ${\tilde{\omega }}^{2}(t)=\Delta ^{-1}(t)\left[ \ddot{\chi }_{{\varvec{u}}}(t)\dot{\chi }_{{\varvec{x}}}(t)-\ddot{\chi }_{{\varvec{x}}}(t)\dot{\chi }_{{\varvec{u}}}(t)\right]$ and $\Delta (t)=\left[ \dot{\chi }_{{\varvec{u}}}(t)\chi _{{\varvec{x}}}(t)-\dot{\chi }_{{\varvec{x}}}(t)\chi _{{\varvec{u}}}(t)\right]$.

Thus, the required FPLE can be read as

$$\begin{aligned}{} & {} \frac{\partial {P}}{\partial {t}} = -{\varvec{u}}\cdot \nabla _{\mathbf{{x}}}P-\left[ \dot{\varvec{p_{x}}}+{\tilde{\beta }}(t)\varvec{p_{x}}+{\tilde{\omega }}^{2}(t)\varvec{q_{x}}\right] .\nonumber \\{} & {} \quad \nabla _{\mathbf{{u}}}P + {\tilde{\omega }}^{2}(t){\varvec{x}}.\nabla _{\mathbf{{u}}}P + {\tilde{\beta }}(t)\nabla _{\mathbf{{u}}}\cdot {\varvec{u}}P \nonumber \\{} & {} \quad + \frac{k_{\text {B}}T}{\omega ^2}\left[ {\tilde{\omega }}^{2}(t)-\omega ^2\right] \nabla _{\mathbf{{u}}}.\nabla _{\mathbf{{x}}}P + \frac{k_{\text {B}}T}{\omega ^2}{\tilde{\beta }}(t) {\nabla _{\mathbf{{u}}}}^{2}P.\nonumber \\ \end{aligned}$$

(A8)

Now, one can check that the distribution function (A2) is a solution of the above equation. Thus, we arrive at the new FPLE with a shortcut way. Shortly, we will compare this equation with its counterpart,i.e., the Markovian case.

For further checking, one may use the condition, ${\varvec{a}}(t)=0$. Then, $\varvec{q_{x}}=\varvec{p_{x}}=0$, as well as $\varvec{G_{1}}(t)=0,$ and the above equation reduces to the standard result [16, 38, 40]

$$\begin{aligned} \frac{\partial {P}}{\partial {t}}= & {} -{\varvec{u}}\cdot \nabla _{\mathbf{{x}}}P + {\tilde{\omega }}^{2}(t){\varvec{x}}.\nabla _{\mathbf{{u}}}P+{\tilde{\beta }}(t)\nabla _{\mathbf{{u}}}\cdot {\varvec{u}}P \nonumber \\{} & {} + \frac{k_{\text {B}}T}{\omega ^2}\left[ {\tilde{\omega }}^{2}(t)-\omega ^2\right] \nabla _{\mathbf{{u}}}.\nabla _{\mathbf{{x}}}P + \frac{k_{\text {B}}T}{\omega ^2}{\tilde{\beta }}(t) {\nabla _{\mathbf{{u}}}}^{2}P. \qquad \quad \end{aligned}$$

(A9)

At the Markovian limit, Eq. (A8) reduces to

$$\begin{aligned} \frac{\partial {P}}{\partial {t}}= & {} -{\varvec{u}}\cdot \nabla _{\mathbf{{x}}}P+\omega ^{2}{\varvec{x}}.\nabla _{\mathbf{{u}}}P+\gamma _{0}\nabla _{\mathbf{{u}}}\cdot {\varvec{u}}P\nonumber \\{} & {} -\varvec{a(t)}\cdot \nabla _{\mathbf{{u}}}P+ \gamma _{0}k_\textrm{B}T{\nabla _{\mathbf{{u}}}}^{2}P. \end{aligned}$$

(A10)

In the absence of the time-dependent force ($\varvec{a(t)}=0$), the above equation becomes the Kramers’ equation for the isotropic harmonic oscillator [1].

Comparing Eq. (A9) with Eq. (A10), we find that as a signature of the time-dependent damping, 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 Eqs. (A9) and (A10), respectively. Here again, it is apparent that the coefficient for the non-Markovian dynamics is time-dependent. The diffusion term in Eq. (A9) with cross derivatives is of purely non-Markovian origin. Its coefficient easily measures the deviation of the dynamics from the Markovian character. Finally, comparing Eq. (A8) with Eq. (A10), we find 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. The first and second terms on the right-hand side of Eq. (A7) correspond to the respective terms in Eq. (10) for the non-Markovian dynamics of time-dependent external force-driven Brownian particle. The remaining term in Eq. (A7) is due to the harmonic force field. Thus, the present method is very simple to use compared to the other methods to identify the modulation. It is to be noted here that the cyclotron frequency becomes time-dependent for the non-Markovian dynamics [40], like the harmonic oscillator case. 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.

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 [39]:

$$\begin{aligned} \dot{{\varvec{u}}}={\varvec{F}}+{\varvec{f}}(t)+{\varvec{a}}(t), \end{aligned}$$

(B1)

where

$$\begin{aligned} {\varvec{F}}=-\int _{0}^{t}\gamma (t-\tau ){\varvec{u}}(\tau ){\text {d}}\tau +\frac{q}{m}({\varvec{u}}\times {\varvec{B}}). \end{aligned}$$

(B2)

${\varvec{B}}=(0,0,B_{z})$, 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 the $x--y$ plane. The related equations of motion in terms of components of velocity are:

$$\begin{aligned} \dot{u_{x}}=-\int _{0}^{t}\gamma (t-\tau )u_{x}(\tau ){\text {d}}\tau +\Omega u_{y}+f_{x}(t)+a_{x}(t) \qquad \quad \end{aligned}$$

(B3)

and

$$\begin{aligned} \dot{u_{y}}=-\int _{0}^{t}\gamma (t-\tau )u_{y}(\tau ){\text {d}}\tau -\Omega u_{x}+f_{y}(t)+a_{y}(t), \nonumber \\ \end{aligned}$$

(B4)

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, ${\varvec{a}}(t)$.

Now the velocity distribution function for the Langevin Eqs. (B3–B4) can be written as

$$\begin{aligned}{} & {} P(u_{x},u_{y},u_{x}(0),u_{y}(0);t)=\left( \frac{1}{2\pi }\right) \left[ \frac{1}{|\varvec{{\mathcal {A}}(t)}|}\right] ^{\frac{1}{2}}\nonumber \\{} & {} \quad \times \exp \left[ -\frac{1}{2}g\dag (t)\varvec{{\mathcal {A}}}^{-1}(t)g(t)\right] \end{aligned}$$

(B5)

with

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

(B6)

where

$$\begin{aligned} g_{1}(t)= & {} u_{x}(t)-\left[ \left( \chi _{1}(t)u_{x}(0)+\chi _{2}(t)u_{y}(0)\right) \right. \nonumber \\{} & {} \left. +\left( \int _{0}^{t}\chi _{1}(\tau )a_{x}(t-\tau ){\text {d}}\tau +\int _{0}^{t}\chi _{2}(\tau )a_{y}(t-\tau ){\text {d}}\tau \right) \right] \nonumber \\= & {} u_{x}(t)-\left[ c_{1}+p_{x}\right] =\int _{0}^{t}\chi _{1}(\tau )f_{x}(t-\tau ){\text {d}}\tau \nonumber \\{} & {} +\int _{0}^{t}\chi _{2}(\tau )f_{y}(t-\tau ){\text {d}}\tau \ \end{aligned}$$

(B7)

and

$$\begin{aligned} g_{2}(t)= & {} u_{y}(t)-\left[ \left( \chi _{1}(t)u_{y}(0)-\chi _{2}(t)u_{x}(0)\right) \right. \nonumber \\{} & {} \left. +\left( \int _{0}^{t}\chi _{1}(\tau )a_{y}(t-\tau ){\text {d}}\tau -\int _{0}^{t}\chi _{2}(\tau )a_{x}(t-\tau ){\text {d}}\tau \right) \right] \nonumber \\= & {} u_{y}(t)-\left[ c_{2}+p_{y}\right] =\int _{0}^{t}\chi _{1}(\tau )f_{y}(t-\tau ){\text {d}}\tau \nonumber \\{} & {} -\int _{0}^{t}\chi _{2}(\tau )f_{x}(t-\tau ){\text {d}}\tau . \end{aligned}$$

(B8)

Here, we have used $c_{1}=\chi _{1}u_{x}(0)+\chi _{2}u_{y}(0)$, $c_{2}=\chi _{1}u_{y}(0)-\chi _{2}u_{x}(0)$, $p_{x}=\int _{0}^{t}\chi _{1}(\tau )a_{x}(t-\tau ){\text {d}}\tau +\int _{0}^{t}\chi _{2}(\tau )a_{y}(t-\tau ){\text {d}}\tau$ and $p_{y}=\int _{0}^{t}\chi _{1}(\tau )a_{y}(t-\tau ){\text {d}}\tau -\int _{0}^{t}\chi _{2}(\tau )a_{x}(t-\tau ){\text {d}}\tau$. The response functions, $\chi _{1}(t)$ and $\chi _{2}(t)$, which appear in the above equations, are inverse Laplace transform of the following relations: ${\tilde{\chi }}_{1}(s)=\frac{\left[ s+{\tilde{\gamma }}(s)\right] }{\left[ s+{\tilde{\gamma }}(s)\right] ^{2}+\Omega ^{2}}$ and ${\tilde{\chi }}_{2}(s)=\frac{\Omega }{\left[ s+{\tilde{\gamma }}(s)\right] ^{2}+\Omega ^{2}}$,

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

We are now in a position to propose the FPLE for the non-Markovian dynamics with Eqs. (B3–B4) [40] as

$$\begin{aligned} \frac{\partial {P}}{\partial {t}}= & {} \beta _{1}\left[ \nabla _{\mathbf{{u}}}.\mathbf{{u}}P\right] +\beta _{2}\left[ \mathbf{{u}}\times \nabla _{\mathbf{{u}}}P\right] _{z}+G_{1}(t)\frac{\partial P}{\partial u_{x}}\nonumber \\{} & {} +G_{2}(t)\frac{\partial P}{\partial u_{y}}+H(t){\nabla _{\mathbf{{u}}}}^2 P, \end{aligned}$$

(B9)

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

Following Sect. 2.1, we determine these coefficients as $\beta _{1}(t)=-\frac{(\chi _{1}\dot{\chi }_{1}+\chi _{2}\dot{\chi }_{2})}{(\chi _{1}^{2}+\chi _{2}^{2})}$, $\beta _{2}(t)=-\frac{(\chi _{2}\dot{\chi _{1}}-\chi _{1}\dot{\chi }_{2})}{(\chi _{1}^{2}+\chi _{2}^{2})}$, $H(t)=\frac{(\chi _{1}^{2}+\chi _{2}^{2})}{2}\frac{\text {d}}{{\text {d}}t}\left[ \frac{A(t)}{(\chi _{1}^{2}+\chi _{2}^{2})}\right]$. The remaining drift coefficients are given by

$$\begin{aligned} G_{1}(t)=-\dot{p_{x}}-\beta _{1}(t)p_{x}+\beta _{2}(t)p_{y}, \end{aligned}$$

(B10)

and

$$\begin{aligned} G_{2}(t)=-\dot{p_{y}}-\beta _{1}(t)p_{y}-\beta _{2}(t)p_{x}. \end{aligned}$$

(B11)

Then Eq. (B9) becomes

$$\begin{aligned} \frac{\partial {P}}{\partial {t}}= & {} \beta _{1}(t)\nabla _{\mathbf{{u}}}.\mathbf{{u}}P-\left[ \mathbf {{\dot{p}}}+\beta _{1}(t)\mathbf{{p}}\right] .\nabla _{\mathbf{{u}}}P - \beta _{2}(t)\left[ \mathbf{{p}}\times \nabla _{\mathbf{{u}}}P\right] _{z} \nonumber \\{} & {} + \beta _{2}(t)\left[ \mathbf{{u}}\times \nabla _{\mathbf{{u}}}P\right] _{z}+H(t){\nabla _{\mathbf{{u}}}}^{2}P, \end{aligned}$$

(B12)

where $\mathbf{{p}}=\left( p_{x},p_{y},0\right)$. This is the required FPLE. Now one can check that the distribution function (B5) 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. [39] using the characteristic function. Now comparing this subsection with Sect. 2 in Ref. [39], 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 demonstrate in IIB.

For further checking, one can show easily that in the absence of time-dependent force, $\varvec{a(t)}=0,$ and the above equation reduces to the following known result [39, 40]:

$$\begin{aligned} \begin{array}{rcl} \dfrac{\partial {P}}{\partial {t}}&{=}&\beta _{1}(t)\left[ \nabla _{\mathbf{{u}}}.\mathbf{{u}}P\right] \end{array}{+}\beta _{2}(t)\left[ \mathbf{{u}}\times \nabla _{\mathbf{{u}}}P\right] _{z}{+}H(t){\nabla _{\mathbf{{u}}}}^{2}P, \nonumber \\ \end{aligned}$$

(B13)

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

Finally, at the Markovian limit, the FPLE (B12) reduces to

$$\begin{aligned} \dfrac{\partial {P}}{\partial {t}} {=} \gamma _{0}\left[ \nabla _{\mathbf{{u}}}.\mathbf{{u}}P\right] -\mathbf{{a}.}\nabla _{\mathbf{{u}}}P {+}\Omega \left[ \mathbf{{u}}\times \nabla _{\mathbf{{u}}}P\right] _{z}{+}\gamma _{0}k_\textrm{B}T{\nabla _{\mathbf{{u}}}}^{2}P. \nonumber \\ \end{aligned}$$

(B14)

In the absence of a time-dependent force field, ${\varvec{a}}(t)=0$, the above equation is the same as that obtained for the Markovian Brownian motion and is described by a charged particle across a magnetic field [98, 99]. Thus, the FPLE (B12) satisfies all possible limiting situations. However, it is to be noted here that comparing the above equation with Eqs. (B12–B13), one may find how the cyclotron frequency and the effect of the time-dependent external force field are modulated in the presence of a non-Markovian thermal bath. Equations (A7, B10–B11) show that the modulation is quite similar to the case of 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. It is to be noted here that sign for the last terms in Eqs. (B10–B11) is different by virtue of the nature of the magnetic force. However, in Sect. 2.2, we present the relevant modulations in the presence of both conservative and non-conservative force fields, respectively.

Solution of Eqs. (31–32) and related quantities

,of Eqs. (31-32) can be written as

$$\begin{aligned} g_{1}(t)= & {} x(t)-\left( A(t)x(0)-B(t)y(0)+C(t)u_{x}(0)+D(t)u_{y}(0)\right) \nonumber \\{} & {} - \left( \int _{0}^{t}H_{0}(t-\tau )a_{x}(\tau ){\text {d}}\tau -\Omega ^{2}\int _{0}^{t}H_{0}^{\prime }(t-\tau )a_{x}(\tau ){\text {d}}\tau \right. \nonumber \\{} & {} \left. +\Omega \int _{0}^{t}H(t-\tau )a_{y}(\tau ){\text {d}}\tau \right) \nonumber \\= & {} x(t)-\left[ c_{1}+q_{x}\right] \nonumber \\= & {} \int _{0}^{t}H_{0}(t-\tau )f_{x}(\tau ){\text {d}}\tau -\Omega ^{2}\int _{0}^{t}H_{0}^{\prime }(t-\tau )f_{x}(\tau ){\text {d}}\tau \nonumber \\{} & {} +\Omega \int _{0}^{t}H(t-\tau )f_{y}(\tau ){\text {d}}\tau , \end{aligned}$$

(C1)

$$\begin{aligned} g_{2}(t)= & {} y(t)-\left( A(t)y(0)+B(t)x(0)+C(t)u_{y}(0)-D(t)u_{x}(0)\right) \nonumber \\{} & {} - \left( \int _{0}^{t}H_{0}(t-\tau )a_{y}(\tau ){\text {d}}\tau -\Omega ^{2}\int _{0}^{t}H_{0}^{\prime }(t-\tau )a_{y}(\tau ){\text {d}}\tau \right. \nonumber \\{} & {} \left. -\Omega \int _{0}^{t}H(t-\tau )a_{x}(\tau ){\text {d}}\tau \right) \nonumber \\= & {} y(t)-\left[ c_{2}+q_{y}\right] \nonumber \\= & {} \int _{0}^{t}H_{0}(t-\tau )f_{y}(\tau ){\text {d}}\tau -\Omega ^{2}\int _{0}^{t}H_{0}^{\prime }(t-\tau )f_{y}(\tau ){\text {d}}\tau \nonumber \\{} & {} -\Omega \int _{0}^{t}H(t-\tau )f_{x}(\tau ){\text {d}}\tau , \end{aligned}$$

(C2)

$$\begin{aligned} g_{3}(t)= & {} u_{x}(t)-\left( {\dot{A}}(t)x(0)-{\dot{B}}(t)y(0)+{\dot{C}}(t)u_{x}(0)+{\dot{D}}(t)u_{y}(0)\right) \nonumber \\{} & {} - \left( \int _{0}^{t}\dot{H_{0}}(t-\tau )a_{x}(\tau ){\text {d}}\tau -\Omega ^{2}\int _{0}^{t}\dot{H_{0}^{\prime }}(t-\tau )a_{x}(\tau ){\text {d}}\tau \right. \nonumber \\{} & {} \left. +\Omega \int _{0}^{t}{\dot{H}}(t-\tau )a_{y}(\tau ){\text {d}}\tau \right) \nonumber \\= & {} u_{x}(t)-\left[ c_{3}+p_{x}\right] \nonumber \\= & {} \int _{0}^{t}\dot{H_{0}}(t-\tau )f_{x}(\tau ){\text {d}}\tau -\Omega ^{2}\int _{0}^{t}\dot{H_{0}^{\prime }}(t-\tau )f_{x}(\tau ){\text {d}}\tau \nonumber \\{} & {} +\Omega \int _{0}^{t}{\dot{H}}(t-\tau )f_{y}(\tau ){\text {d}}\tau , \end{aligned}$$

(C3)

$$\begin{aligned} g_{4}(t)= & {} u_{y}(t)-\left( {\dot{A}}(t)y(0)+{\dot{B}}(t)x(0)+{\dot{C}}(t)u_{y}(0)-{\dot{D}}(t)u_{x}(0)\right) \nonumber \\{} & {} - \left( \int _{0}^{t}\dot{H_{0}}(t-\tau )a_{y}(\tau ){\text {d}}\tau -\Omega ^{2}\int _{0}^{t}\dot{H_{0}^{\prime }}(t-\tau )a_{y}(\tau ){\text {d}}\tau \right. \nonumber \\{} & {} \left. -\Omega \int _{0}^{t}{\dot{H}}(t-\tau )a_{x}(\tau ){\text {d}}\tau \right) \nonumber \\= & {} u_{y}(t)-\left[ c_{4}+p_{y}\right] \nonumber \\= & {} \int _{0}^{t}\dot{H_{0}}(t-\tau )f_{y}(\tau ){\text {d}}\tau -\Omega ^{2}\int _{0}^{t}\dot{H_{0}^{\prime }}(t-\tau )f_{y}(\tau ){\text {d}}\tau \nonumber \\{} & {} -\Omega \int _{0}^{t}{\dot{H}}(t-\tau )f_{x}(\tau ){\text {d}}\tau , \end{aligned}$$

(C4)

where $A\equiv A(t)=\chi _{0}(t)+\Omega ^{2}\omega ^{2}\chi (t)$, $B\equiv B(t)=\Omega \omega ^{2}H^{\prime }(t)$, $C\equiv C(t)=H_{0}(t)-\Omega ^{2}H_{0}^{\prime }(t)$, $D\equiv D(t)=\Omega H(t)$ with $\chi _{0}(t)=1-\omega ^{2}\int _{0}^{t}H_{0}(\tau ){\text {d}}\tau$ and $\chi (t)=\int _{0}^{t}H_{0}^{\prime }(\tau ){\text {d}}\tau$. These relations imply 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 ${\tilde{H}}_{0}(s)=\frac{1}{s^{2}+s{\tilde{\gamma }}(s)+\omega ^{2}}$, $\tilde{H^{\prime }}(s)=\frac{1}{(s^{2}+s{\tilde{\gamma }}(s)+\omega ^{2})^{2}+(\Omega s)^{2}}$ and $\tilde{H_{00}}(s)=\frac{1}{(s^{2}+s{\tilde{\gamma }}(s)+\omega ^{2})[(s^{2}+s{\tilde{\gamma }}(s)+\omega ^{2})^{2}+(\Omega s)^{2}]}$.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. (C1–C4)) can be represented by the matrix, $\varvec{{\mathcal {A}}}^{\prime }(t)$ with $A_{ij}^{\prime }=\langle g_{i}(t)g_{j}(t)\rangle$. Using Eqs. (C1–C4), one can read the matrix $\varvec{{\mathcal {A}}}^{\prime }(t)$ as

$$\begin{aligned} \varvec{{\mathcal {A}}}^{\prime }=\begin{bmatrix}A_{1} &{} 0 &{} A_{3} &{} A_{4}\\ 0 &{} A_{1} &{} -A_{4} &{} A_{3}\\ A_{3} &{} -A_{4} &{} A_{2} &{} 0\\ A_{4} &{} A_{3} &{} 0 &{} A_{2}, \end{bmatrix} \end{aligned}$$

(C5)

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

$$\begin{aligned} {\varvec{{\mathcal {A}}}^{\prime }}^{-1}(t){} & {} =\frac{adj(\varvec{{\mathcal {A}}}^{\prime }(t))}{|\varvec{{\mathcal {A}}}^{\prime }(t)|}=\frac{1}{(A_{1}A_{2}-A_{3}^{2}-A_{4}^{2})}\nonumber \\{} & {} \quad \times \begin{bmatrix}A_{2} &{} 0 &{} -A_{3} &{} -A_{4}\\ 0 &{} A_{2} &{} A_{4} &{} -A_{3}\\ -A_{3} &{} A_{4} &{} A_{1} &{} 0\\ -A_{4} &{} -A_{3} &{} 0 &{} A_{1} \end{bmatrix}. \end{aligned}$$

(C6)

Time-dependent coefficients of the FPLE (35)

Following Ref. [88], we use the distribution function (33) in Eq. (35) and then get the time-dependent coefficients as $H_{1}=\frac{\left[ -{\ddot{A}}{\tilde{a}}_{x}(t)-{\ddot{B}}{\tilde{a}}_{y}(t)-{\ddot{C}}{\tilde{a}}_{v_{x}}(t)-{\ddot{D}}{\tilde{a}}_{v_{y}}(t))\right] }{\Delta _{m}}$, $H_{2}=\frac{\left[ {\ddot{A}}{\tilde{b}}_{x}(t)-{\ddot{B}}{\tilde{b}}_{y}(t)-{\ddot{C}}{\tilde{b}}_{v_{x}}(t)-{\ddot{D}}{\tilde{b}}_{v_{y}}(t))\right] }{\Delta _{m}}$, $H_{3}=\frac{\left[ -{\ddot{A}}{\tilde{d}}_{x}(t)+{\ddot{B}}{\tilde{d}}_{y}(t)-{\ddot{C}}{\tilde{d}}_{v_{x}}(t)+{\ddot{D}}{\tilde{d}}_{v_{y}}(t))\right] }{\Delta _{m}}$, $H_{4}=\frac{\left[ -{\ddot{A}}{\tilde{c}}_{x}(t)+{\ddot{B}}{\tilde{c}}_{y}(t)+{\ddot{C}}{\tilde{c}}_{v_{x}}(t)-{\ddot{D}}{\tilde{c}}_{v_{y}}(t))\right] }{\Delta _{m}}$, $H_{5}=\left[ \dot{A_{4}}+H_{2}A_{4}+H_{3}A_{3}-H_{4}A_{1}\right]$, $H_{6}=\left[ \dot{A_{3}}-A_{2}-H_{3}A_{4}+H_{1}A_{1}+H_{2}A_{3}\right]$, and $H_{7}=\frac{1}{2}\left[ \dot{A_{2}}+2H_{1}A_{3}+2H_{2}A_{2}-2H_{4}A_{4}\right]$. Here, we have used $\Delta _{m}=(A^{2}+B^{2})({\dot{C}}^{2}+{\dot{D}}^{2})+(C^{2}+D^{2})({\dot{A}}^{2}+{\dot{B}}^{2})-2(AC-BD)({\dot{A}}{\dot{C}}-{\dot{B}}{\dot{D}})-2(AD+BC)({\dot{A}}{\dot{D}}+{\dot{B}}{\dot{C}})$, ${\tilde{a}}_{x}(t)=A({\dot{C}}^{2}+{\dot{D}}^{2})-C({\dot{A}}{\dot{C}}-{\dot{B}}{\dot{D}})-D({\dot{A}}{\dot{D}}+{\dot{B}}{\dot{C}})$, ${\tilde{c}}_{x}(t)=B({\dot{C}}^{2}+{\dot{D}}^{2})+D({\dot{A}}{\dot{C}}-{\dot{B}}{\dot{D}})-C({\dot{A}}{\dot{D}}+{\dot{B}}{\dot{C}})$, ${\tilde{b}}_{x}(t)=B(C{\dot{D}}-{\dot{C}}D)+C(A{\dot{C}}-{\dot{A}}C)+D(A{\dot{D}}-D{\dot{A}})$, ${\tilde{d}}_{x}(t)=B(C{\dot{C}}+D{\dot{D}})-C(A{\dot{D}}+C{\dot{B}})+D(A{\dot{C}}-D{\dot{B}})$, ${\tilde{a}}_{y}(t)=B({\dot{C}}^{2}+{\dot{D}}^{2})+D({\dot{A}}{\dot{C}}-{\dot{B}}{\dot{D}})-C({\dot{A}}{\dot{D}}+{\dot{B}}{\dot{C}})$, ${\tilde{c}}_{y}(t)=A({\dot{C}}^{2}+{\dot{D}}^{2})-C({\dot{A}}{\dot{C}}-{\dot{B}}{\dot{D}})-D({\dot{A}}{\dot{D}}+{\dot{B}}{\dot{C}})$, ${\tilde{b}}_{y}(t)=A(C{\dot{D}}-{\dot{C}}D)-C(B{\dot{C}}-{\dot{B}}C)-D(B{\dot{D}}-{\dot{B}}D)$, ${\tilde{d}}_{y}(t)=A(C{\dot{C}}+{\dot{D}}D)+C(B{\dot{D}}-{\dot{A}}C)-D(B{\dot{C}}+{\dot{A}}D)$, ${\tilde{a}}_{v_{x}}(t)=C({\dot{A}}^{2}+{\dot{B}}^{2})-A({\dot{A}}{\dot{C}}-{\dot{B}}{\dot{D}})-B({\dot{A}}{\dot{D}}+{\dot{B}}{\dot{C}})$, ${\tilde{c}}_{v_{x}}(t)=D({\dot{A}}^{2}+{\dot{B}}^{2})+B({\dot{A}}{\dot{C}}-{\dot{B}}{\dot{D}})-A({\dot{A}}{\dot{D}}+{\dot{B}}{\dot{C}})$, ${\tilde{b}}_{v_{x}}(t)=A(A{\dot{C}}-{\dot{A}}C)+B(B{\dot{C}}-{\dot{B}}C)-D(A{\dot{B}}-{\dot{A}}B)$, ${\tilde{d}}_{v_{x}}(t)=A(A{\dot{D}}+{\dot{B}}C)+B(B{\dot{D}}-{\dot{A}}C)-D(A{\dot{A}}+{\dot{B}}B)$, ${\tilde{a}}_{v_{y}}(t)=D({\dot{A}}^{2}+{\dot{B}}^{2})+B({\dot{A}}{\dot{C}}-{\dot{B}}{\dot{D}})-A({\dot{A}}{\dot{D}}+{\dot{B}}{\dot{C}})$, ${\tilde{c}}_{v_{y}}(t)=C({\dot{A}}^{2}+{\dot{B}}^{2})-A({\dot{A}}{\dot{C}}-{\dot{B}}{\dot{D}})-B({\dot{A}}{\dot{D}}+{\dot{B}}{\dot{C}})$, ${\tilde{b}}_{v_{y}}(t)=A(A{\dot{D}}-{\dot{A}}D)+B(B{\dot{D}}-{\dot{B}}D)+C(A{\dot{B}}-{\dot{A}}B)$, and ${\tilde{d}}_{v_{y}}(t)=A(A{\dot{C}}-{\dot{B}}D)+B(B{\dot{C}}+{\dot{A}}D)-C(A{\dot{A}}+{\dot{B}}B)$. It is to be noted here that all the time-dependent coefficients do not depend on $c_{1}$, $c_{2}$, $c_{3}$ and $c_{4}$ as in the previous cases with ${\varvec{a}}(t)=0$. Then following the earlier cases, we have

$$\begin{aligned} G_{1}=-\dot{p_{x}}-H_{1}q_{x}-H_{2}p_{x}+H_{3}p_{y}-H_{4}q_{y}, \end{aligned}$$

(D1)

$$\begin{aligned} G_{2}=-\dot{p_{y}}-H_{1}q_{y}-H_{2}p_{y}-H_{3}p_{x}+H_{4}q_{x}. \end{aligned}$$

(D2)

The last terms in Eqs. (D1–D1) are due to non-Markovian dynamics-induced fictitious electric field. Equations (38–39) imply that the sign of these terms may be opposite to each other. Thus, in the modulation process, the individual force puts its own identity.