Asymptotic Behavior of the Velocity Distribution of Driven Inelastic Gas with Scalar Velocities: Analytical Results

Abstract

We determine the asymptotic behavior of the tails of the steady state velocity distribution of a homogeneously driven granular gas comprising of particles having a scalar velocity. A pair of particles undergo binary inelastic collisions at a rate that is proportional to a power of their relative velocity. At constant rate, each particle is driven by multiplying its velocity by a factor $-r_w$ and adding a stochastic noise. When $r_w <1$, we show analytically that the tails of the velocity distribution are primarily determined by the noise statistics, and determine analytically all the parameters characterizing the velocity distribution in terms of the parameters characterizing the stochastic noise. Surprisingly, we find logarithmic corrections to the leading stretched exponential behavior. When $r_w=1$, we show that for a range of distributions of the noise, inter-particle collisions lead to a universal tail for the velocity distribution.

Appendices

Appendix A: Large Moments of the Velocity Distribution

We determine the behavior of large moments when the velocity distribution has the asymptotic form:

$$\begin{aligned} P(v) \sim e^{-a |v|^\beta +\varPsi (v)},~~|v|\rightarrow \infty ,~a>0. \end{aligned}$$

(83)

where $v^{-\beta } \varPsi (|v|)\rightarrow 0$ for $|v|\rightarrow \infty$. The aim is to determine the moment $\langle |v_1-v_2|^\delta v_1^{2n-k}v_2^k\rangle$ for large n and k. In the limit $k \propto n$, we change variables to $k=xn$ with $0\le x\le 1$, to obtain

$$\begin{aligned} \langle |v_1-v_2|^\delta v_1^{2n-k}v_2^{k}\rangle\sim & {} \int dv_1dv_2|v_1-v_2|^\delta v_1^{(2-x)n}v_2^{xn}\nonumber \\&\times e^{-a|v_1|^\beta -a|v_2|^\beta +\varPsi (v_1)+\varPsi (v_2)} \end{aligned}$$

(84)

Rescaling the integration variables $v_1$, and $v_2$ as $v_1=n^{1/\beta }t_1$ and $v_2=n^{1/\beta }t_2$, Eq. (84) simplifies to

$$\begin{aligned} \langle |v_1-v_2|^\delta v_1^{2n-k}v_2^{k}\rangle&\sim n^{\frac{2+\delta + 2 n}{\beta }} \int dt_1dt_2e^{n[f(t_1)+ g(t_2)]}\nonumber \\&\quad \times |t_1-t_2|^\delta e^{\varPsi (n^{1/\beta }t_1)+\varPsi (n^{1/\beta }t_2)} \end{aligned}$$

(85)

where,

$$\begin{aligned} \begin{aligned} f(t_1)&=-at_1^\beta +(2-x)\ln t_1,\\ g(t_2)&=-at_2^\beta +x\ln t_2. \end{aligned} \end{aligned}$$

(86)

The integral in Eq. (85) may be evaluated using the saddle point approximation by maximizing $f(t_1)$ and $g(t_2)$ with respect to $t_1$ and $t_2$. Setting $df(t_1)/dt_1=0$ and $dg(t_2)/dt_2=0$, we obtain the solution $t_1^*$ and $t_2^*$ to be

$$\begin{aligned} t_1^*&=\left( \frac{2-x}{a\beta }\right) ^\frac{1}{\beta }, \end{aligned}$$

(87)

$$\begin{aligned} t_2^*&=\left( \frac{x}{a\beta }\right) ^\frac{1}{\beta }, \end{aligned}$$

(88)

and

$$\begin{aligned} \begin{aligned} f(t_1^*)&=\frac{2-x}{\beta }\ln \left( \frac{2-x}{ae\beta }\right) ,\\ g(t_2^*)&=\frac{x}{\beta }\ln \left( \frac{x}{ae\beta }\right) . \end{aligned} \end{aligned}$$

(89)

Doing the saddle point integration in Eq. (85) obtain

$$\begin{aligned} \langle |v_1-v_2|^\delta v_1^{2n-k}v_2^k\rangle&\sim \frac{n^\frac{2+\delta }{\beta }}{n}\left( \frac{n}{2ae\beta }\right) ^\frac{2n}{\beta }\nonumber \\&\quad \times e^{n\left[ \frac{2-x}{\beta }\ln (2-x)+ \frac{x}{\beta }\ln x\right] } e^{\varPsi (n^{1/\beta }t_1^*)+\varPsi (n^{1/\beta }t_2^*)}, \end{aligned}$$

(90)

where the extra factor of n appears in the denominator of the left hand side of Eq. (90) because of the two-dimensional saddle point integration.

In a similar way one can find the asymptotic form for $\langle |v_1-v_2|^\delta v_1^{2n}\rangle$ for large n. Consider the equality,

$$\begin{aligned} \langle |v-v_1|^\delta v^{2n}\rangle =\int dvdv_1P(v)P(v_1)|v-v_1|^\delta v^{2n}. \end{aligned}$$

(91)

As we are interested in the large n, the moments are dominated by values of |v|. In this limit it is possible to approximate, $|v-v_1|\approx |v|$, and Eq. (91) reduces to

$$\begin{aligned} \langle |v-v_1|^\delta v^{2n}\rangle \sim \int dvP(v)|v|^{2n+\delta } . \end{aligned}$$

(92)

Doing a transformation, $v=n^{1/\beta }t$ as before, we obtain

$$\begin{aligned} \langle |v-v_1|^\delta v^{2n}\rangle \sim n^{\frac{1+\delta }{\beta }}\int dte^{nf(t)+\varPsi (n^{1/\beta }t)}n^{2n/\beta } \end{aligned}$$

(93)

where,

$$\begin{aligned} f(t)=-at^\beta +2\ln t \end{aligned}$$

(94)

Doing a saddle point integration with respect to t, by maximizing f(t), one obtains,

$$\begin{aligned} \langle |v{-}v_1|^\delta v^{2n}\rangle \approx n^{\frac{1+\delta }{\beta }}\sqrt{\frac{2\pi }{f''(t^*)n}}{t^*}^\delta \left( \frac{2n}{ae\beta }\right) ^{\frac{2n}{\beta }} e^{\varPsi (n^{\frac{1}{\beta }}t^*)}~~ \end{aligned}$$

(95)

where we have substituted, the maximal value of f(t),

$$\begin{aligned} f(t^*)=\frac{2}{\beta }\ln \left( \frac{2}{ae\beta }\right) , \end{aligned}$$

(96)

and

$$\begin{aligned} t^*=\left( \frac{2}{a\beta }\right) ^\frac{1}{\beta }. \end{aligned}$$

(97)

The same calculation may be used to determine the asymptotic behavior of the noise distribution. Considering the noise distribution with the form as in Eq. (15),

$$\begin{aligned} \varPhi (\eta )\sim |\eta |^{\tilde{\chi }}e^{-b|\eta |^\gamma } \text {for} |\eta |^2 \gg \langle \eta ^2\rangle _\varPhi , \end{aligned}$$

(98)

then

$$\begin{aligned} N_{2n} = \langle \eta ^{2n}\rangle \approx n^{\frac{1+\tilde{\chi }}{\gamma }}\sqrt{\frac{2\pi }{g''(y^*)n}}{y^*}^{\tilde{\chi }} \left[ \frac{2 n}{b e \gamma }\right] ^{\frac{2 n}{\gamma }}. \end{aligned}$$

(99)

where,

$$\begin{aligned} g(y)=-by^\gamma +2\ln y \end{aligned}$$

(100)

and

$$\begin{aligned} y^*=\left( \frac{2}{b\gamma }\right) ^\frac{1}{\gamma }. \end{aligned}$$

(101)

Appendix B: Taylor Expansion of the Terms in Eq. (11) Arising from Driving

In this appendix, we analyse the terms [the last two terms] of Eq. (11) arising from driving—which we denote by I—for $r_w=1$:

$$\begin{aligned} I=-\lambda _d P(v) + \lambda _d \int \int d\eta dv_1 \varPhi (\eta ) P(v_1) \delta \left[ - v_1+\eta -v \right] , \end{aligned}$$

(102)

by Taylor expanding the term in the integrand, and using the solution for the velocity distribution that we have obtained in the paper to identify the most dominant term in the expansion. Integrating over $v_1$, we obtain

$$\begin{aligned} I=-\lambda _d P(v) + \lambda _d \int d\eta \varPhi (\eta ) P(v - \eta ). \end{aligned}$$

(103)

Taylor expanding $P(v-\eta )$ about $\eta =0$ and integrating over $\eta$, Eq. (103) reduces to

$$\begin{aligned} I=\lambda _d \sum _{k=1}^\infty \frac{\langle \eta ^{2 k} \rangle }{(2 k)!} \frac{d^{2k}}{d v^{2k}} P(v)\equiv \sum _{k=1}^\infty t_k. \end{aligned}$$

(104)

We now determine the dominant term in this expansion, given that P(v) has the asymptotic behaviour as given in Eqs. (3)–(7).

From Eqs. (3)–(7), the velocity distribution has the asymptotic form

$$\begin{aligned} \ln P(v) \sim -a |v|^\beta (\ln |v|)^\theta , \quad |v| \rightarrow \infty , \end{aligned}$$

(105)

where $\theta >0$ for $\gamma >1$, $\delta >0$, and $\theta =0$ otherwise. Thus,

$$\begin{aligned} \frac{d^{2k}}{d v^{2k}} P(v) \approx \left[ a \beta v^{\beta -1} (\ln |v|)^\theta \right] ^{2 k} P(v), \quad |v| \rightarrow \infty . \end{aligned}$$

(106)

The asymptotic form for the moments of $\eta$ is given in Eq. (22). Using these asymptotic forms, we obtain the ratio of successive terms in Eq. (104) to be

$$\begin{aligned} \frac{t_{k+1}}{t_k} \approx \left( \frac{2 }{b \gamma }\right) ^{\frac{2}{\gamma }} \frac{\left[ a \beta v^{\beta -1} (\ln |v|)^\theta \right] ^{2} }{4} k^{\frac{2(1-\gamma )}{\gamma }}. \end{aligned}$$

(107)

We now analyse Eq. (107) for different cases.

Case I$\delta \le 0$: From Eq. (3), we know that $\theta =0$ and $\beta =\min \left[ \frac{2 + \delta }{2}, \gamma \right]$. When $\delta <0$, then $\beta <1$. Thus, for large v, the ratio in Eq. (107) is much smaller than 1 for small k. Also, for $\gamma >1$, the ratio decreases with k. Thus, the truncation at the first term is valid, and the expression for $\beta$ may be obtained from the continuum equations. However, when $\gamma <1$, it is possible that the expansion may break down as there could be contribution from large derivatives. Indeed, from comparing with our analytic solution, for $\gamma < 1+\delta /2$, the Taylor expansion about small $\eta$ breaks down. When $\delta =0$, then $\beta =1$. For $\gamma >1$, the most dominant term in the expansion, determined by $t_{k+1}/t_k = 1$, is a $k^*$ that is independent of v. Then, it is straightforward to show that $\beta =1$ is a consistent solution. Thus, while the dominant term in the expansion is not the first term, the result obtained from the continuum equations do not change.

Case II$\delta > 0$: Now, for $\gamma >1$, the solution of$t_{k^*+1}/t_{k^*} =1$ is given by $k^* \propto (\ln |v|)^{\frac{\theta \gamma }{\gamma -1}}$. Now, $k^*$ depends on v, and it is clear that truncating the Taylor expansion at the first term will lead to wrong results. In fact, truncating at the first term gives $\beta = (2+\delta )/2$, while we have derived, without making any approximations, $\beta =1$.