Asymptotic Derivation of Langevin-like Equation with Non-Gaussian Noise and Its Analytical Solution

Abstract

We asymptotically derive a non-linear Langevin-like equation with non-Gaussian white noise for a wide class of stochastic systems associated with multiple stochastic environments, by developing the expansion method in our previous paper (Kanazawa et al. in Phys Rev Lett 114:090601–090606, 2015). We further obtain a full-order asymptotic formula of the steady distribution function in terms of a large friction coefficient for a non-Gaussian Langevin equation with an arbitrary non-linear frictional force. The first-order truncation of our formula leads to the independent-kick model and the higher-order correction terms directly correspond to the multiple-kicks effect during relaxation. We introduce a diagrammatic representation to illustrate the physical meaning of the high-order correction terms. As a demonstration, we apply our formula to a granular motor under Coulombic friction and get good agreement with our numerical simulations.

Appendices

Appendix 1: Relation to the Non-equilibrium Steady State: Granular Rotor Under Viscous Friction

Fig. 15

(Color online) a Schematic of rotor associated with the granular and molecular rarefied gases. The rotor is composed of the two cuboids ($h\times w \times l$), and is driven by the collisional impulses by the granular and molecular gas whose velocity distribution functions (VDFs) are $\phi _g(v)$ and $\phi _m(v)$, respectively. The granular and molecular gases are so dilute that their collisional impacts $\hat{F}_g(t;\omega )$ and $\hat{F}_m(t;\omega )$ are described by the Boltzmann–Lorentz models with the transition rates $W_g(\omega ; y)$ and $W_m(\omega ; y)$, respectively. b Schematic of the collisional rules between the rotor and a granular (or molecular) particle

We here study the relation between our formulation and the non-equilibrium steady state through the example of the granular motor under viscous friction. We consider a rotor placed in the granular and molecular rarefied gases characterized by isotropic velocity distributions $\phi _g(|\mathbf {v}|)$ and $\phi _m(|\mathbf {v}|)$ (see Fig. 15a). The rotor is cuboid with height h, width w, and depth l. The masses of the rotor, the granular particle, and the molecular particle are M, $m_g$, and $m_m$, respectively. For simplicity, we assume that all the restitution coefficients are equal to 1 and $l=0$. If the granular and molecular gases are sufficiently dilute (i.e., their density $\rho _g$ and $\rho _m$ are sufficiently small), the dynamics of the rotor are governed by the Boltzmann–Lorentz equation as:

$$\begin{aligned} \frac{\partial P(\omega ,t)}{\partial t} = \sum _{i=g,m} \int _{-\infty }^\infty dy[P(\omega -y,t)W_i(\omega -y;y)-P(\omega ,t)W_i(\omega ;y)], \end{aligned}$$

(127)

$$\begin{aligned} W_i(\omega ;y) = \rho _i h \int _{0}^{2w} dx\int _{-\infty }^\infty d\mathbf {v} \phi _i(|\mathbf {v}|)\Theta (\Delta \mathbf {V}(x)\cdot \mathbf {n}(x))|\Delta \mathbf {V}(x)\cdot \mathbf {n}(x)|\delta (y-\Delta \omega _i(x)),\nonumber \\ \end{aligned}$$

(128)

where x is the coordinate along the cuboid (see Fig. 15b), $\mathbf {n}(x)$ is the normal unit vector to the surface at the point x, and we introduce the inertia radius $R_I\equiv \sqrt{I/M}$ and

$$\begin{aligned} \mathbf {V}(x) \equiv \omega \mathbf {e}_z\times \mathbf {r}(x), \quad g(x)\equiv \frac{\mathbf {r}(x)\cdot \mathbf {t}(x)}{R_I}, \quad \mathbf {t}(x) \equiv \mathbf {e}_z \times \mathbf {n}(x), \quad r_{i} \equiv \frac{m_i}{M}, \end{aligned}$$

(129)

$$\begin{aligned} \Delta \mathbf {V}(x) \equiv \mathbf {V}(x)-\mathbf {v}, \quad \Delta \omega _i(x) \equiv \frac{\Delta \mathbf {V}(x)\cdot \mathbf {n}(x)}{R_I}\frac{2r_i g(x)}{1+r_i g^2(x)}. \end{aligned}$$

(130)

We here assume that the granular mass ratio $\varepsilon \equiv r_g= m_g/M$ is small. Furthermore, we make the following three assumptions:

• (A1) The masses of the rotor, granular, and molecular particles satisfy the relations $M\gg m_g \gg m_m$. In other words, the mass ratio $r_i$ is scaled as

  $$\begin{aligned} \frac{r_m}{r_g} =O(\varepsilon ) \Longleftrightarrow r_m=\varepsilon c_r r_g=\varepsilon ^2 c_r, \end{aligned}$$

  (131)

  where $c_r$ is a dimensionless constant independent of $\varepsilon$.

• (A2) The density of the granular gas is much smaller than that of the molecular gas as $\rho _m \gg \rho _g$. In other words, $\rho _g$ is scaled as

  $$\begin{aligned} \frac{\rho _g}{\rho _m} = O(\varepsilon ^2) \Longleftrightarrow \rho _g = \varepsilon ^2 c_\rho \rho _m, \end{aligned}$$

  (132)

  where $c_\rho$ is a dimensionless constant independent of $\varepsilon$. This assumption implies that the collision frequency of the granular particles is much lower than that of the molecular particles.

• (A3) The velocity distributions $\phi _i(|\mathbf {v}|)$ are Maxwellian forms characterized by temperatures $T_i$ for $i=g,m$:

  $$\begin{aligned} \phi _i(|\mathbf {v}|) = \left( \frac{m_i}{2\pi T_i}\right) ^{3/2}\exp {\left[ -\frac{m_i|\mathbf {v}|^2}{2T_i}\right] }. \end{aligned}$$

  (133)

  Furthermore, the granular temperature $T_g$ is much higher than the molecular temperature $T_m$ as $T_g \gg T_m$. In other words, $T_m$ is scaled with an $\varepsilon$-independent dimensionless constant $c_T$ as

  $$\begin{aligned} \frac{T_m}{T_g} = O(\varepsilon ) \Longleftrightarrow T_m = \varepsilon c_T T_g. \end{aligned}$$

  (134)

Under the assumptions (A1)–(A3), we use the Kramers–Moyal expansion for the molecular gas:

$$\begin{aligned}&\int _{-\infty }^\infty dy[P(\omega -y,t)W_m(\omega -y;y)-P(\omega ,t)W_m(\omega ;y)]\nonumber \\&=\sum _{n=1}^\infty \frac{(-1)^n\varepsilon ^{2n}}{n!}\frac{\partial ^n }{\partial \omega ^n}\left[ K_n(\omega )P(\omega ,t)\right] \end{aligned}$$

(135)

with the scaled Kramers–Moyal coefficient

$$\begin{aligned} \begin{array}{c} \displaystyle K_n (\omega ) \equiv \int dy y^nW_m(\omega ;y)\\ \quad =\rho _m h\int _0^{2w}\!\!dx \int _{-\infty }^\infty d\mathbf {v} \phi _m(|\mathbf {v}|)\Theta (\Delta \mathbf {V}\cdot \mathbf {n})|\Delta \mathbf {V}\cdot \mathbf {n}|\frac{(2c_r g(x) \Delta \mathbf {V}\cdot \mathbf {n})^n}{R_I^n(1+\varepsilon ^2 c_rg^2(x))^n}, \end{array} \end{aligned}$$

(136)

where $\phi _m(|\mathbf {v}|)$ is independent of $\varepsilon$ as $\phi _m(|\mathbf {v}|) = (c_r m_g/2\pi c_TT_g)^{3/2}\exp {[-c_r m_g|\mathbf {v}|^2/2c_T T_g]}$. The scaled Kramers–Moyal coefficients are expanded as

$$\begin{aligned} K_n (\omega ) = \sum _{k=0}^\infty \frac{K_{n;(k)}^*}{k!}\omega ^{k}, \end{aligned}$$

(137)

where $K_{1;(0)}^*=0$ and $K_{1;(1)}^*\ne 0$. Introducing the scaled variables

$$\begin{aligned} \tau \equiv \varepsilon ^2t,\quad \Omega \equiv \frac{\omega }{\varepsilon }, \end{aligned}$$

(138)

we obtain the scaled master equation (127) as

$$\begin{aligned} \frac{\partial \mathcal {P}(\Omega ,\tau )}{\partial \tau }&= \left[ \sum _{k=0}^\infty \frac{\varepsilon ^k}{k!}\left\{ -\frac{K_{1;(k+1)}^*}{k+1}\frac{\partial }{\partial \Omega }\Omega ^{k+1}+\frac{K_{2;(k)}^*}{2}\frac{\partial }{\partial \Omega }\Omega ^{k} \right\} \right. \nonumber \\&\left. \quad \, +\, \sum _{n=3}^\infty \sum _{k=0}^\infty \frac{(-1)^n\varepsilon ^{n+k-2}K_{n;(k)}^*}{n!k!}\frac{\partial ^n }{\partial \Omega ^n}\Omega ^k\right] \mathcal {P}(\Omega ,\tau )\nonumber \\&\quad +\,\int _{-\infty }^\infty d\mathcal {Y}[\mathcal {P}(\Omega -\mathcal {Y},\tau )\tilde{W}_g(\Omega -\mathcal {Y};\mathcal {Y})-\mathcal {P}(\Omega ,\tau )\tilde{W}_g(\Omega ;\mathcal {Y})], \end{aligned}$$

(139)

$$\begin{aligned} \tilde{W}_g(\Omega ;\mathcal {Y})= & {} c_\rho \rho _m h \int _{0}^{2w} dx\int _{-\infty }^\infty d\mathbf {v} \phi _g(|\mathbf {v}|)\Theta (\Delta {\tilde{\mathbf {V}}}(x)\cdot \mathbf {n}(x))|\Delta {\tilde{\mathbf {V}}}(x)\nonumber \\&\cdot \mathbf {n}(x)|\delta (\mathcal {Y}-\Delta \Omega _g(x)), \end{aligned}$$

(140)

where $\phi _g(|\mathbf {v}|)$ is independent of $\varepsilon$ as $\phi _g(|\mathbf {v}|) = (m_g/2\pi T_g)^{3/2}\exp {[-m_g|\mathbf {v}|^2/2 T_g]}$ and we introduce

$$\begin{aligned} {\tilde{\mathbf {V}}}(x) \equiv \Omega \mathbf {e}_z\times \mathbf {r}(x), \quad \Delta {\tilde{\mathbf {V}}}(x) \equiv \varepsilon {\tilde{\mathbf {V}}}(x)-\mathbf {v}, \quad \Delta \Omega _g(x) \equiv \frac{\Delta {\tilde{\mathbf {V}}}(x)\cdot \mathbf {n}(x)}{R_I}\frac{2g(x)}{1+\varepsilon g^2(x)}.\nonumber \\ \end{aligned}$$

(141)

In the limit $\varepsilon \rightarrow 0$, Eq. (139) is reduced to

$$\begin{aligned} \frac{\partial \mathcal {P}(\Omega , \tau )}{\partial \tau } = \gamma \left[ \frac{\partial }{\partial \Omega }\Omega + \frac{\mathcal {T}_m}{\mathcal {I}}\frac{\partial ^2 }{\partial \Omega ^2}\right] \mathcal {P}(\Omega , \tau ) + \int _{-\infty }^\infty \mathcal {W}(\mathcal {Y})[\mathcal {P}(\Omega -\mathcal {Y}, \tau )-\mathcal {P}(\Omega , \tau )],\nonumber \\ \end{aligned}$$

(142)

$$\begin{aligned} \mathcal {W}(\mathcal {Y}) = 2c_\rho \rho _m h \int _{0}^{w} dx\int _{-\infty }^\infty d\mathbf {v} \phi _g(|\mathbf {v}|)\Theta (- \mathbf {v}\cdot \mathbf {n}(x))|\mathbf {v}\cdot \mathbf {n}(x)|\delta (\mathcal {Y}-\Delta \Omega (x)), \end{aligned}$$

(143)

where we have used

$$\begin{aligned} K^*_{1;(1)} = - \frac{hw^3\rho _m}{3R_I^2}\sqrt{\frac{2c_r c_T T_g}{\pi m_g}} \equiv -\gamma ,\quad K_{2;(0)}^* = \frac{2hw^3 \rho _m c_T T_g}{3R_I^4m_g}\sqrt{\frac{2c_r c_T T_g}{\pi m_g}} =\frac{2\gamma \mathcal {T}_m}{\mathcal {I}}\nonumber \\ \end{aligned}$$

(144)

with $\mathcal {T}_m\equiv c_T T_g$ and $\mathcal {I}\equiv m_gR_I^2$. Equation (142) is equivalent to the non-Gaussian Langevin equation

$$\begin{aligned} \frac{d\hat{\Omega }}{d\tau } = -\gamma \hat{\Omega }+ \sqrt{2\gamma \mathcal {T}_m/\mathcal {I}}\hat{\xi }_\mathrm{G} + \hat{\xi }_\mathrm{NG}, \end{aligned}$$

(145)

where $\hat{\xi }_\mathrm{NG}$ is characterized by the transition rate $\mathcal {W}(\mathcal {Y})$. As can be seen in the assumption (A3), the non-Gaussian Langevin equation (145) is only valid for system connected with two reservoirs characterized by extremely different temperatures. If there is no temperature difference (i.e., the system is in equilibrium condition as $T_g=T_m$), the non-Gaussian Langevin equation (145) does not appear. We also note that there exists a energy current from the granular to the molecular gas.

Appendix 2: Check of the Solution (56) of the Integral Equation (55)

In this appendix, we check that the solution (56) satisfies the integral equation (55) on the condition that $\tilde{a}_n(0)=0$. We note that Eq. (55) belongs to the class of the first-kind Fredholm integral equations with convolution kernels [86]. Substituting the solution (56) into the left hand-side of Eq. (55), we obtain

$$\begin{aligned}&\frac{1}{2\pi }\int _{-\infty }^\infty du \tilde{f}(s-u)\tilde{a}_{n+1}(u)\nonumber \\&\quad = \frac{1}{2\pi }\int _{-\infty }^\infty du \tilde{f}(s-u)\mathcal {I}[u;\tilde{a}_n(s')]\nonumber \\&\quad = \frac{1}{(2\pi )^2}\int _{-\infty }^\infty du \tilde{f}(s-u)\int _{-\infty }^\infty \frac{d\mathcal {V}(e^{iu\mathcal {V}}-1)}{f(\mathcal {V})}\int _{-\infty }^\infty ds'e^{-is'\mathcal {V}}\frac{\Phi (s')}{is'}\tilde{a}_n(s')\nonumber \\&\quad = \frac{1}{(2\pi )^2}\int _{-\infty }^\infty \frac{d\mathcal {V}}{f(\mathcal {V})}\int _{-\infty }^\infty du \tilde{f}(s-u)(e^{iu\mathcal {V}}-1)\int _{-\infty }^\infty ds'e^{-is'\mathcal {V}}\frac{\Phi (s')}{is'}\tilde{a}_n(s')\nonumber \\&\quad = \frac{1}{2\pi }\int _{-\infty }^\infty \frac{d\mathcal {V}}{f(\mathcal {V})}f(\mathcal {V})e^{is\mathcal {V}}\int _{-\infty }^\infty ds'e^{-is'\mathcal {V}}\frac{\Phi (s')}{is'}\tilde{a}_n(s')\nonumber \\&\quad = \int _{-\infty }^\infty ds'\delta (s-s')\frac{\Phi (s')}{is'}\tilde{a}_n(s') = \frac{\Phi (s)}{is}\tilde{a}_n(s), \end{aligned}$$

(146)

where we have used the relation $\int _{-\infty }^\infty du\tilde{f}(s-u)=f(0)=0$ in the third line. We note that the solution (56) satisfies the condition for the conservation of the probability $\tilde{a}_n(0)=0$. Equation (56) is then the solution of the integral equation (55).

Appendix 3: Derivation of the Asymptotic Tail (77) for the Cubic Friction

We here check that the explicit form of the asymptotic tail (77) for the cubic friction. We first assume that $|d\tilde{P}/ds| \gg |d^3\tilde{P}/ds^3|$ for $s\rightarrow \infty$. Using the method of dominant balance, we obtain

$$\begin{aligned} \frac{d\tilde{P}(s)}{ds}\simeq & {} \frac{\mu \Phi (s)}{s}\tilde{P}(s) \Longrightarrow \tilde{P}(s) \simeq \exp \left[ \mu \int _0^s \frac{\Phi (s')}{s'}\right] \nonumber \\= & {} \exp \left[ 2\mu \int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^\mathcal {Y} d\mathcal {V}\frac{\cos {s\mathcal {V}}-1}{\mathcal {V}}\right] , \end{aligned}$$

(147)

where we have used the relation $\int _0^s ds'(\cos {s'\mathcal {Y}}-1)/s' = \int _0^{\mathcal {Y}} d\mathcal {V}(\cos {s\mathcal {V}}-1)/\mathcal {V}$. Note that $d\tilde{P}(s)/ds$ and $d^3\tilde{P}(s)/ds^3$ decay for $s\rightarrow \infty$ as

$$\begin{aligned} \frac{1}{\tilde{P}(s)}\frac{d\tilde{P}(s)}{ds} \sim -\frac{\mu \lambda ^*}{s} + o(s^{-1}),\quad \frac{1}{\tilde{P}}\frac{d^3\tilde{P}(s)}{ds^3} \sim -\frac{\mu ^3\lambda ^{*3}}{s} + o\big (s^{-1}\big ), \end{aligned}$$

(148)

where we have introduced $\lambda ^* \equiv 2\int _{0}^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})$ and used the relation $\lim _{s\rightarrow \infty }\Phi (s) = -\lambda ^*$. Equation (148) ensures the consistency of the assumption $|d\tilde{P}/ds| \gg |d^3\tilde{P}/ds^3|$ for $s\rightarrow \infty$. We note that the solution (147) asymptotically behaves as

$$\begin{aligned} \tilde{P}(s) \simeq \exp \left[ -2\mu \int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\mathrm{Cin}(s\mathcal {Y})\right] \sim \exp \left[ -2\mu \int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\log {s}\right] =|s|^{-\mu \lambda ^*},\nonumber \\ \end{aligned}$$

(149)

where we have used the asymptotic form of the cosine integral $\mathrm{Cin}(x)\equiv \int _0^x dt(1-\cos {t})/t \sim \log {x}$ for $x\rightarrow \infty$.

Appendix 4: Check of the Renormalized Solution (84)

In this appendix, we check whether the postulated expression (84) satisfies the ordinary differential equation (76) by the direct substitution. For simplicity, we assume that $\mathcal {W}(\mathcal {Y})$ is an $L^2$-function, where $\Phi (s)$ is an bounded function as $|\Phi (s)| \le \lambda ^*$ with $\lambda ^* \equiv 2\int _{0}^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})$. We note that $\Phi (s)/s$ is also a bounded function because $\Phi (s)/s$ is regular at $s=0$ as $\lim _{s\rightarrow 0}\Phi (s)/s = 0$ due to the symmetry $\Phi (s)=\Phi (-s)$. We also note that $\Phi (s)/s$ behaves as $\Phi (s)/s = O(s^{-1})$ in the limit $s\rightarrow \infty$. The differential equation (76) then has only two singular points at $s=\pm \infty$. We here introduce the following quantities:

$$\begin{aligned} Q_1(s)&\equiv 2\int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^\mathcal {Y} d\mathcal {V}\frac{\cos {s\mathcal {V}}-1}{\mathcal {V}(1+\mathcal {V}^2)},\nonumber \\ Q_2(s)&\equiv \frac{dQ_1}{ds}= -2\int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^\mathcal {Y} d\mathcal {V}\frac{\sin {s\mathcal {V}}}{1+\mathcal {V}^2}, \end{aligned}$$

(150)

$$\begin{aligned} Q_3(s)&\equiv \frac{dQ_2}{ds} = -2\int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^\mathcal {Y} d\mathcal {V}\frac{\mathcal {V}\cos {s\mathcal {V}}}{1+\mathcal {V}^2},\nonumber \\ Q_4(s)&\equiv \frac{dQ_3}{ds}= 2\int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^\mathcal {Y} d\mathcal {V}\frac{\mathcal {V}^2\sin {s\mathcal {V}}}{1+\mathcal {V}^2}. \end{aligned}$$

(151)

For these quantities, the following relations hold:

$$\begin{aligned} \frac{d}{ds}e^{\mu Q_1(s)} = \mu Q_2 e^{\mu Q_1(s)},\quad \frac{d^3}{ds^3}e^{\mu Q_1(s)} = \left[ \mu Q_4 + 3\mu ^2 Q_2Q_3+\mu ^3 Q_2^3\right] e^{\mu Q_1(s)}.\nonumber \\ \end{aligned}$$

(152)

Then, we obtain

$$\begin{aligned}&\left( \frac{d}{ds}-\frac{d^3}{ds^3}\right) e^{\mu Q_1(s)} \nonumber \\&\quad \,= -\left( 2\mu \int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^{\mathcal {Y}}\sin {s\mathcal {V}}\right) e^{\mu Q_1(s)} + \left[ 3\mu ^2 Q_2Q_3+\mu ^3 Q_2^3\right] e^{\mu Q_1(s)}\nonumber \\&\quad \,= \left( 2\mu \int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})(\cos {s\mathcal {Y}}-1)\right) e^{\mu Q_1(s)} + \left[ 3\mu ^2 Q_2Q_3+\mu ^3 Q_2^3\right] e^{\mu Q_1(s)}\nonumber \\&\quad \,= \frac{\mu \Phi (s)}{s}e^{\mu Q_1(s)} + \left[ 3\mu ^2 Q_2Q_3+\mu ^3 Q_2^3\right] e^{\mu Q_1(s)}. \end{aligned}$$

(153)

Note that $Q_2$ and $Q_3$ are bounded as

$$\begin{aligned} |Q_2(s)| \le 2\int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^\mathcal {Y} d\mathcal {V}\left| \frac{\sin {s\mathcal {V}}}{1+\mathcal {V}^2}\right| \le 2E,\quad |Q_3| \le 2\int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^\mathcal {Y}\nonumber \\ d\mathcal {V}\left| \frac{\mathcal {V}^2\sin {s\mathcal {V}}}{1+\mathcal {V}^2}\right| \le E,\qquad \end{aligned}$$

(154)

where $E\equiv \int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\mathcal {Y}>0$. Furthermore, $\Phi (s)/s$, $Q_2$, and $Q_3$ decay for $s\rightarrow \infty$ as

$$\begin{aligned} \frac{\Phi (s)}{s}&= \int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\frac{\cos {s\mathcal {Y}}-1}{s} \simeq - \frac{\lambda ^*}{2s} +O\big (s^{-2}\big ), \end{aligned}$$

(155)

$$\begin{aligned} Q_2(s)&= -2\int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^\mathcal {Y} d\mathcal {V}\frac{\sin {s\mathcal {V}}}{1+\mathcal {V}^2} \simeq -2\int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^\infty d\mathcal {V}\frac{\sin {s\mathcal {V}}}{1+\mathcal {V}^2}\nonumber \\&= -\frac{\lambda ^*}{s} + O\big (s^{-3}\big ),\end{aligned}$$

(156)

$$\begin{aligned} Q_3(s)&= -2\int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^\mathcal {Y} d\mathcal {V}\frac{\mathcal {V}\cos {s\mathcal {V}}}{1+\mathcal {V}^2}\nonumber \\&\simeq -2\int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\int _0^\infty d\mathcal {V}\frac{\mathcal {V}\cos {s\mathcal {V}}}{1+\mathcal {V}^2} = \frac{\lambda ^*}{s^2} + O\big (s^{-4}\big ), \end{aligned}$$

(157)

where we have used the Riemann-Lebesgue lemma as $\lim _{s\rightarrow \infty }\int _0^\infty \mathcal {W}(\mathcal {Y}) \cos {s\mathcal {V}}=0$. This implies that the second term on the rhs of Eq. (153) is negligible compared with the first term for $s\rightarrow \infty$ and that $\tilde{P}(s)=e^{Q_1(s)/\gamma }$ uniformly satisfies Eq. (76) up to the first-order. We also note asymptotic relations as

$$\begin{aligned}&\left| e^{-\mu Q_1(s)}\left[ \left( \frac{d}{ds}-\frac{d^3}{ds^3}\right) e^{\mu Q_1(s)} - \frac{\mu \Phi (s)}{s}e^{\mu Q_1(s)}\right] \right| \le 6\mu ^2E^2 + 8\mu ^3 E^3,\end{aligned}$$

(158)

$$\begin{aligned}&\left| e^{-\mu Q_1(s)}\left[ \left( \frac{d}{ds}-\frac{d^3}{ds^3}\right) e^{\mu Q_1(s)} - \frac{\mu \Phi (s)}{s}e^{\mu Q_1(s)}\right] \right| \nonumber \\&= \frac{-3\mu ^2\lambda ^{2*}}{ s^3} + \frac{\mu ^3\lambda ^{2*}}{ s^2} + o\big (s^{-4}\big )\quad (s\rightarrow \infty ) \end{aligned}$$

(159)

Appendix 5: Heuristic Derivation of the Non-smooth Coulombic Friction from a Smooth Friction

We heuristically derive the non-smooth property of Coulombic friction from a smooth friction in this appendix. For simplicity, we set $l=0$ and consider the case where the solid friction is given by a hyperbolic smooth friction

$$\begin{aligned} f(\omega ) = \gamma \tanh \left( \frac{\omega }{\omega _\mathrm{C}}\right) , \end{aligned}$$

(160)

where $\omega _\mathrm{C}$ is the characteristic angular velocity scale for the solid friction. Note that this form of a friction is used to analyze the non-smoothness of Coulombic friction in Ref. [75] because it recovers Coulomb friction as

$$\begin{aligned} f(\omega ) = \gamma \mathrm{\>sgn}(\omega ) \big (1-e^{-2|\omega |/\omega _\mathrm{C}}\big )\big (1+e^{-2|\omega |/\omega _\mathrm{C}}\big )^{-1} \simeq \gamma \mathrm{\>sgn}(\omega ) \quad (|\omega | \gg \omega _\mathrm{C}). \end{aligned}$$

(161)

We here assume the following two scalings. The first scaling is the OCR scaling

$$\begin{aligned} \beta ^{-1} \equiv \frac{\tau _\mathrm{R}}{\tau _\mathrm{C}} \sim 1 \Longleftrightarrow \gamma = \varepsilon \gamma ', \end{aligned}$$

(162)

which implies that the relaxation time originating from Coulombic friction is equivalent to the typical collision interval. The second scaling is for the typical angular velocities:

$$\begin{aligned} \alpha ^{-1} \equiv \frac{\omega _\mathrm{G}}{\omega _\mathrm{C}} = O\big (\varepsilon ^{-1/2}\big ) \gg 1 \Longleftrightarrow \omega _\mathrm{C}/\varepsilon = \alpha \Omega _w \propto \varepsilon ^{1/2}, \end{aligned}$$

(163)

where $\omega _\mathrm{G}\equiv \varepsilon \Omega _w$ is the typical angular velocity jump by granular impulses. This scaling implies that the typical angular velocity during relaxation $\omega _\mathrm{G}$ is much larger than $\omega _\mathrm{C}$, and the hyperbolic friction can be approximated as

$$\begin{aligned} f(\omega ) = \varepsilon \gamma ' \tanh \left( \frac{\Omega }{\omega _\mathrm{C}/\varepsilon }\right) \simeq \varepsilon \gamma '\mathrm{\>sgn}(\Omega ) \quad (|\Omega | \ne 0). \end{aligned}$$

(164)

On these conditions, the formulation in Sect. 4.2 can be heuristically validated. Note that the emergence of the non-smooth Coulombic friction is studied more rigorously for the Gaussian noise in Ref. [75]. We also note that the derivation of Coulombic friciton and the estimation of the friction coefficient are addressed in Ref. [87].

Appendix 6: Derivation of the Granular Langevin Equation in the FCL

In this appendix, we derive the granular Langevin equation for the FCL regime on the basis of the parallel formulation in Sect. 2.3. Let us consider the case where $l=0$ and the FCL scaling $\beta ^{-1} = O(\varepsilon ^{-1/2}) \gg 1$ is satisfied. We then introduce the scaled friction $\tilde{\gamma }'\equiv \beta \varepsilon ^{-1/2} \rho S v_0^2/R_I$, which implies

$$\begin{aligned} \gamma = \varepsilon ^{3/2}\tilde{\gamma }'. \end{aligned}$$

(165)

We further introduce the FCL scaling variables as follows:

$$\begin{aligned} \tilde{\Omega } \equiv \frac{\omega }{\sqrt{\varepsilon }}, \quad \tau \equiv \varepsilon t. \end{aligned}$$

(166)

The Kramers–Moyal expansion of Eq. (106) is given by

$$\begin{aligned} \frac{\partial \mathcal {P}(\tilde{\Omega },\tau )}{\partial \tau }&= \left[ \frac{\partial }{\partial \tilde{\Omega }}\tilde{\gamma }'\mathrm{\>sgn}(\tilde{\Omega }) + \sum _{n=1}^\infty \frac{(-1)^n\varepsilon ^{n/2-1}}{n!}\frac{\partial ^n}{\partial \tilde{\Omega }^n}\mathcal {K}_n(\tilde{\Omega })\right] \mathcal {P}(\tilde{\Omega },\tau )\nonumber \\&= \left[ \frac{\partial }{\partial \tilde{\Omega }}\tilde{\gamma }'\mathrm{\>sgn}(\tilde{\Omega }) - \frac{\partial }{\partial \tilde{\Omega }}\mathcal {K}^*_{1;(1)}\tilde{\Omega } + \frac{\mathcal {K}^*_{2;(0)}}{2}\frac{\partial ^2}{\partial \tilde{\Omega }^2}\right] \mathcal {P}(\tilde{\Omega },\tau ) + O\big (\varepsilon ^{1/2}\big ), \end{aligned}$$

(167)

where we have introduced the Kramers–Moyal coefficient and its expansion

$$\begin{aligned} \mathcal {K}_n(\tilde{\Omega })&= \rho h \int _0^{2w} dx \int _{-\infty }^\infty d\mathbf {v} \phi (|\mathbf {v}|)\Theta (\Delta \mathbf {V} \cdot \mathbf {n})|\Delta \mathbf {V}\cdot \mathbf {n}|\frac{[(1+e)(\Delta V\cdot \mathbf {n})g]^n}{R_I^n (1+\varepsilon g^2)^n},\nonumber \\ \mathcal {K}_n(\tilde{\Omega })&= \sum _{k=0}^\infty \frac{\mathcal {K}^*_{n;(k)}}{k!}\tilde{\Omega }^k \end{aligned}$$

(168)

with $\mathcal {K}^*_{1;(0)}=0$. We then obtain the Gaussian Langevin equation in the FCL

$$\begin{aligned} \frac{d\tilde{\Omega }}{d\tau } = -\gamma \mathrm{\>sgn}(\tilde{\Omega }) - \gamma _g \tilde{\Omega } + \sqrt{\Gamma _g}\hat{\xi }_\mathrm{{G}}, \end{aligned}$$

(169)

where $\gamma _g\equiv -\mathcal {K}^*_{1;(1)} = (\pi \rho h (1+e)w^3 /3R_I^2)\int _0^\infty dv v^3 \phi (v)$ and $\Gamma _g \equiv \mathcal {K}^*_{2;(0)} = (\pi \rho h (1+e)^2 w^3/12R_I^4)\int _0^\infty dvv^5\phi (v)$. This result is consistent with the theoretical and experimental results in Refs. [32–34, 84].

Appendix 7: Cumulant Generating Function of the Granular Noise (116)

In this appendix, we derive the explicit form of the cumulant generating function of the granular noise (116). The cumulant generating function $\Phi (s)$ can be transformed as

$$\begin{aligned} \Phi (s)&= \int _{-\infty }^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\big (e^{is\mathcal {Y}}-1\big )\nonumber \\&=\, \rho h\int _{-\infty }^\infty d\mathcal {Y} \int _{0}^{2(l+w)}dx \int _{-\infty }^\infty d\mathbf {v} \big (e^{is\mathcal {Y}}-1\big )\phi (|\mathbf {v}|)\Theta (-\mathbf {v}\cdot \mathbf {n}(x))|\mathbf {v}\cdot \mathbf {n}(x)|\delta \nonumber \\&\quad \left[ \mathcal {Y}+(1+e)\frac{\mathbf {v}\cdot \mathbf {n}(x)}{R_I}g(x)\right] \nonumber \\&=\, \rho h\int _{0}^{2(l+w)}dx \int _{-\infty }^\infty d\mathbf {v} \phi (|\mathbf {v}|)\Theta (-\mathbf {v}\cdot \mathbf {n}(x))|\mathbf {v}\cdot \mathbf {n}(x)|\big (e^{-is(1+e)(\mathbf {v} \cdot \mathbf {n}(x))g(x)/R_I}-1\big ). \end{aligned}$$

(170)

Introducing the representation of the polar coordinate system $(v,\theta ',\psi )$, we obtain

$$\begin{aligned} \Phi (s)&\!=\! \rho h\int _{0}^{2(l+w)}\!\!\!\! dx \int _0^\infty \!\!\!\!dv \int _0^{2\pi }\!\!\!\!d\theta ' \int _0^\pi \!\!\!d\psi v^2\sin {\psi } \phi (v)\Theta (-v\cos {\psi })|v\nonumber \\&\quad \times \cos {\psi }|\big (e^{-is(1+e)v\cos {\psi }g(x)/R_I}-1\big )\nonumber \\&\!=\! -\rho h\int _{0}^{2(l+w)}\!\!\!\! dx \int _0^\infty \!\!\!\!dv \int _0^{2\pi }\!\!\!\!d\theta ' \int _{\pi /2}^\pi \!\!\!d\psi v^3 \phi (v)\sin {\psi }\cos {\psi }\nonumber \\&\quad \times \big (e^{-i[s(1+e)vg(x)/R_I]\cos {\psi }}\!-\!1\big )\nonumber \\&\!=\! 4\pi \rho h\int _{0}^{(l+w)}\!\!\!\! dx \int _0^\infty \!\!\!\!dv \int _0^1 d\chi v^3\phi (v)\chi \big (e^{is(1+e)vg(x)\chi /R_I}\!-\!1\big )\nonumber \\&\!=\! \Phi _l(s) + \Phi _w(s), \end{aligned}$$

(171)

where we have introduced $\chi = -\cos \psi$ and

$$\begin{aligned} \Phi _l (s)&= 4\pi \rho h\int _{0}^{l} \>\>dx \int _0^\infty \!\!\!\!dv \int _0^1 d\chi v^3\phi (v)\chi \big (e^{is(1+e)vg(x)\chi /R_I}-1\big ), \end{aligned}$$

(172)

$$\begin{aligned} \Phi _w (s)&= 4\pi \rho h\int _{l}^{(l+w)}\!\!\!\!\!\!\!\! dx \int _0^\infty \!\!\!\!dv \int _0^1 d\chi v^3\phi (v)\chi \big (e^{is(1+e)vg(x)\chi /R_I}-1\big ). \end{aligned}$$

(173)

Substituting $g(x)=(x-l/2)/R_I$ for $0\le x\le l$ into Eq. (172), we obtain

$$\begin{aligned} \Phi _l (s)&= 4\pi \rho h\int _{0}^{l}dx \int _0^\infty \!\!\!\!dv \int _0^1 d\chi v^3\phi (v)\chi \big (e^{is(1+e)v(x-l/2)\chi /R^2_I}-1\big ) \nonumber \\&= 4\pi \rho h\int _0^\infty \!\!\!\!dv v^3\phi (v) \int _0^1 d\chi \chi \left[ \frac{2\sin {\frac{s(1+e)v\chi l}{2R_I^2}}}{s(1+e)v\chi /R_I^2}- l\right] \nonumber \\&= 4\pi \rho h\int _0^\infty \!\!\!\!dv v^3\phi (v) \left[ \frac{1-\cos {\frac{s(1+e)v l}{2R_I^2}}}{[s(1+e)v /2R_I^2]^2l}\!-\! \frac{l}{2}\right] \nonumber \\&= -\frac{16\pi \rho h R_I^4}{ls^2(1+e)^2}\int _0^\infty \!\!\!\!dv v\phi (v) \left[ \cos {\frac{s(1+e)v l}{2R_I^2}} -1 + \frac{s^2(1+e)^2l^2v^2}{8R_I^4}\right] . \end{aligned}$$

(174)

We similarly obtain

$$\begin{aligned} \Phi _w (s) = -\frac{16\pi \rho h R_I^4}{ws^2(1+e)^2}\int _0^\infty \!\!\!\!dv v\phi (v) \left[ \cos {\frac{s(1+e)v w}{2R_I^2}} -1 + \frac{s^2(1+e)^2w^2v^2}{8R_I^4}\right] .\qquad \end{aligned}$$

(175)

Appendix 8: First Order Solution of the Angular Velocity’s Distribution for the Granular Motor

We here write the explicit derivation of Eq. (121). From Eqs. (73) and (107), we obtain

$$\begin{aligned} \mathcal {P}_\mathrm{SS} (\Omega )&= \left[ 1-\frac{c_1}{\tilde{\gamma }}\right] \delta (\Omega ) + \frac{ \rho h}{\tilde{\gamma }}\int _0^\infty d\mathcal {Y}{\varvec{1}}_{[-\mathcal {Y},\mathcal {Y}]}(\Omega )\int _0^{2(l+w)}dx\nonumber \\&\quad \times \int d\mathbf {v}\phi (|\mathbf {v}|)\Theta (-\mathbf {v}\cdot \mathbf {n}(x))|\mathbf {v}\cdot \mathbf {n}(x)|\delta (\mathcal {Y}-\Delta \Omega (x))\nonumber \\&= \left[ 1-\frac{c_1}{\tilde{\gamma }}\right] \delta (\Omega ) + \frac{2\rho h}{\tilde{\gamma }}\int _0^\infty d\mathcal {Y}{\varvec{1}}_{[-\mathcal {Y},\mathcal {Y}]}(\Omega )\int _0^{ (l+w)}dx\nonumber \\&\quad \times \int d\mathbf {v}\phi (|\mathbf {v}|)\Theta (-\mathbf {v}\cdot \mathbf {n}(x))|\mathbf {v}\cdot \mathbf {n}(x)|\delta (\mathcal {Y}-\Delta \Omega (x)), \end{aligned}$$

(176)

where we have introduced $c_1 = \int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\mathcal {Y}$. Here we calculate the following integral:

$$\begin{aligned}&\int _0^\infty d\mathcal {Y}{\varvec{1}}_{[-\mathcal {Y},\mathcal {Y}]}(\Omega )\int _0^{l}dx \int d\mathbf {v}\phi (|\mathbf {v}|)\Theta (-\mathbf {v}\cdot \mathbf {n}(x))|\mathbf {v}\cdot \mathbf {n}(x)|\delta (\mathcal {Y}-\Delta \Omega (x))\nonumber \\&\quad =\int _0^\infty d\mathcal {Y}{\varvec{1}}_{[-\mathcal {Y},\mathcal {Y}]}(\Omega )\int _0^{l}dx \int _0^\infty \!\!\!\!dv\int _0^{2\pi }\!\!\!\!d\theta '\int _0^\pi \!\!\!\!d\psi v^3 \sin \psi (-\cos \psi ) \phi (v)\Theta (-v\cos {\psi })\delta \nonumber \\&\qquad \left[ \mathcal {Y}+\frac{(1+e)(x-l/2)v\cos \psi }{R_I^2}\right] \nonumber \\&\quad =2\pi \int _{l/2}^{l}dx \int _0^\infty dv\int _{0}^1 d\chi \chi v^3 \phi (v){\varvec{1}}_{[-\mathcal {Y}^{\dagger }(x,v,\chi ),\mathcal {Y}^{\dagger }(x,v,\chi )]}(\Omega ), \end{aligned}$$

(177)

where we have introduced the polar coordinate $(v,\theta ',\psi )$, the flight distance $\mathcal {Y}^\dagger (x,v,\psi ) \equiv (1+e)(x-l/2)\chi v/R_I^2$, and the variable transformation $\chi = -\cos \psi$, and have used $g(x)=(x-l/2)/R_I$ for $0\le x \le l$. We remark that

$$\begin{aligned} |\Omega | \le \mathcal {Y}^{\dagger }(x,v,\chi ) \Longleftrightarrow \frac{R_I^2 |\Omega |}{(1+e)(x-l/2)\chi }\le v. \end{aligned}$$

(178)

We then rewrite the rhs of Eq. (177) into the following form:

$$\begin{aligned}&2\pi \int _{l/2}^{l}dx \int _0^\infty dv\int _{0}^1 d\chi \chi v^3 \phi (v){\varvec{1}}_{[-\mathcal {Y}^{\dagger }(x,v,\chi ),\mathcal {Y}^{\dagger }(x,v,\chi )]}(\Omega )\nonumber \\&\quad =2\pi \int _0^{l/2} dx' \int _0^1 d\chi \chi \int _{R_I^2|\Omega |/(1+e)x'\chi }^\infty \!\!dv v^3 \phi (v)\nonumber \\&\quad =\frac{\pi l}{2}\int _{2R_I^2|\Omega |/(1+e)l}^\infty dv v\phi (v)\left[ v-\frac{2R_I^2|\Omega |}{(1+e)l}\right] ^2, \end{aligned}$$

(179)

where we have introduced the variable transformation $x'=x-l/2$ and used the identity for an arbitrary positive number c as

$$\begin{aligned} \int _0^{l/2}dx' \int _0^1 d\chi \chi \int _{c/\chi x'}^{\infty } dv v^3\phi (v) = \frac{l}{4}\int _{2c/l}^\infty dv v\phi (v)\left[ v-\frac{2c}{l}\right] ^2. \end{aligned}$$

(180)

Similarly, we obtain

$$\begin{aligned}&\int _0^\infty d\mathcal {Y}{\varvec{1}}_{[-\mathcal {Y},\mathcal {Y}]}(\mathcal {V})\int _{l}^{(l+w)}dx \int d\mathbf {v}\phi (|\mathbf {v}|)\Theta (-\mathbf {v}\cdot \mathbf {n}(x))|\mathbf {v}\cdot \mathbf {n}(x)|\delta (\mathcal {Y}-\Delta \Omega (x))\nonumber \\&\quad =\frac{\pi w}{2}\int _{2R_I^2|\Omega |/(1+e)w}^\infty dv v\phi (v)\left[ v-\frac{2R_I^2|\Omega |}{(1+e)w}\right] ^2. \end{aligned}$$

(181)

We then obtain

$$\begin{aligned}&\frac{\rho h}{\tilde{\gamma }}\int _0^\infty d\mathcal {Y}{\varvec{1}}_{[-\mathcal {Y},\mathcal {Y}]}(\Omega )\int _0^{2(l+w)}dx \int d\mathbf {v}\phi (|\mathbf {v}|)\Theta (-\mathbf {v}\cdot \mathbf {n}(x))\delta (\mathcal {Y}-\Delta \Omega (x))\nonumber \\&\quad =\frac{\pi \rho hl}{\tilde{\gamma }}\int _{|\Omega |/F_l}^\infty \!\!\!\!\!\!\!dv v\phi (v)\left[ v-\frac{|\Omega |}{F_l}\right] ^2 + \frac{\pi \rho hw}{\tilde{\gamma }}\int _{|\Omega |/F_w}^\infty \!\!\!\!\!\!\!dv v\phi (v)\left[ v-\frac{|\Omega |}{F_w}\right] ^2, \end{aligned}$$

(182)

where we have used $F_p\equiv p(1+e)/2R_I^2$ for an arbitrary real number p. From Eqs. (176) and (182), we obtain Eq. (121). We also obtain the explicit form of $c_1$ as follows:

$$\begin{aligned} c_1= & {} 2\int _0^\infty d\mathcal {Y}\mathcal {W}(\mathcal {Y})\mathcal {Y} = 4\rho h\int _0^\infty d\mathcal {Y}\mathcal {Y}\int _0^{(l+w)} dx \nonumber \\&\times \int _{-\infty }^{\infty } d\mathbf {v} \phi (|\mathbf {v}|)\Theta (-\mathbf {v}\cdot \mathbf {n}(x))\delta (\mathcal {Y}-\Delta \Omega (x)). \end{aligned}$$

(183)

We here calculate the following integral as

$$\begin{aligned}&\int _0^\infty d\mathcal {Y}\mathcal {Y}\int _0^{l} dx \int _{-\infty }^{\infty } d\mathbf {v} \phi (|\mathbf {v}|)\Theta (-\mathbf {v}\cdot \mathbf {n}(x))|\mathbf {v}\cdot \mathbf {n}(x)|\delta (\mathcal {Y}-\Delta \Omega (x))\nonumber \\&\quad =2\pi \int _0^\infty d\mathcal {Y}\mathcal {Y}\int _0^{l} dx \int _0^{\infty } dvv^2 \int _{0}^{\pi }d\psi \sin \psi \phi (v)\Theta (-v\cos {\psi })|v\cos \psi |\delta \nonumber \\&\qquad \left( \mathcal {Y}-\frac{(1+e)(x-l/2)v\cos \psi }{R_I^2}\right) \nonumber \\&\quad =\frac{2\pi (1+e)}{R_I^2}\int _0^{l/2} dx'x' \int _0^{\infty } dv v^4\phi (v)\int _{0}^{1}d\chi \chi ^2 = \frac{\pi (1+e)l^2}{12R_I^2}\int _0^\infty dv v^4\phi (v). \end{aligned}$$

(184)

Similarly, we obtain

$$\begin{aligned}&\int _0^\infty d\mathcal {Y}\mathcal {Y}\int _{l}^{l+w} dx \int _{-\infty }^{\infty } d\mathbf {v} \phi (|\mathbf {v}|)\Theta (-\mathbf {v}\cdot \mathbf {n}(x))|\mathbf {v}\cdot \mathbf {n}(x)|\delta (\mathcal {Y}-\Delta \Omega (x)) \nonumber \\&\quad =\frac{\pi (1+e)w^2}{12R_I^2}\int _0^\infty dv v^4\phi (v). \end{aligned}$$

(185)

Equations (183), (184) and (185) lead to Eq. (122).