Homogeneous Cooling State

Abstract

This chapter deals with the problem of the so-called homogeneous cooling state (namely, a homogeneous state where granular temperature monotonically decays in time) for mono- and multicomponent granular gases. Unlike ordinary or classical gases, the Maxwell–Boltzmann velocity distribution is not a solution to the Boltzmann kinetic equation and the exact form of this solution is still unknown. For long times, however, the kinetic equation admits a scaling solution whose form can be approximately obtained by considering the leading terms in a Sonine (Laguerre) polynomial expansion. A new and surprising result (compared to its ordinary gas counterpart) is found for granular mixtures: the well-known energy equipartition theorem is broken for freely cooling systems.

Appendix A

In this Appendix the expression (2.32) for $\mu _2$ by taking $a_3=0$ for the sake of simplicity is derived. In this case, Eq. (2.20) for $p=1$ gives

$$\begin{aligned} \mu _{2}= & {} \frac{1}{2}\pi ^{-d}\int {\mathrm {d}}\mathbf{c}_{1}\int {\mathrm {d}}\mathbf{c}_{2}\left\{ 1+a_2\left[ S_2(c_1^2)+S_2(c_2^2)\right] \right\} {\mathrm {e}}^{-(c_1^2+c_2^2)}\nonumber \\&\times \int {\mathrm {d}}\widehat{\varvec{\sigma }} \varTheta (\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12}^*)(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*) \left( c_1^{2}+c_2^{2}-c_1^{'2}-c_2^{'2}\right) , \end{aligned}$$

(2.128)

where only linear terms in $a_2$ have been considered. Since $\mathbf {c}_{1}'=\mathbf {c}_{1}- (\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12}^*)\widehat{\varvec{\sigma }}$ and $\mathbf {c}_{2}'=\mathbf {c}_{2}+(\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12}^*)\widehat{\varvec{\sigma }}$, then

$$\begin{aligned} c_1^{2}+c_2^{2}-c_1'^{2}-c_2'^{2}=\frac{1-\alpha ^2}{2}(\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12}^*)^2. \end{aligned}$$

(2.129)

After performing the angular integral, the collisional moment $\mu _2$ reduces to

$$\begin{aligned} \mu _{2}=\frac{1}{4}\frac{\pi ^{-(d+1)/2}}{\varGamma \left( \frac{d+3}{2}\right) } (1-\alpha ^2)\int {\mathrm {d}}\mathbf{c}_{1}\int {\mathrm {d}}\mathbf{c}_{2}\left[ 1+2 a_2S_2(c_1^2)\right] g_{12}^{*3}\;{\mathrm {e}}^{-(c_1^2+c_2^2)}, \end{aligned}$$

(2.130)

where use has been made of the result (1.177) for $k=3$ and the symmetry of the integrand with respect to the change of particle indices 1 and 2 has been accounted for. Equation (2.130) can be rewritten as

$$\begin{aligned} \mu _2=\frac{1}{4}\frac{\pi ^{-(d+1)/2}}{\varGamma \left( \frac{d+3}{2}\right) } (1-\alpha ^2)\left\{ I(1)+a_2\left[ I''(1)+(d+2)I'(1)+\frac{d(d+2)}{4}I(1)\right] \right\} , \end{aligned}$$

(2.131)

with

$$\begin{aligned} I(\epsilon )= \int {\mathrm {d}}\mathbf{c}_{1}\int {\mathrm {d}}\mathbf{c}_{2}\; g_{12}^{*3}\;{\mathrm {e}}^{-(\epsilon c_1^2+c_2^2)} \end{aligned}$$

(2.132)

and the primes denote differentiation with respect to $\epsilon$, namely, $I'(1)=(\partial I/\partial \epsilon )_{\epsilon =1}$ and $I''(1)=(\partial ^2 I/\partial \epsilon ^2)_{\epsilon =1}$. The integral $I(\epsilon )$ can be performed by the change of variables

$$\begin{aligned} \mathbf {x}=\mathbf {c}_1-\mathbf {c}_2, \quad \mathbf {y}=\epsilon \mathbf {c}_1+\mathbf {c}_2, \end{aligned}$$

(2.133)

with the Jacobian $(1+\epsilon )^{-d}$. According to Eq. (2.133), $\mathbf {c}_1$ and $\mathbf {c}_2$ can be expressed in terms of $\mathbf {x}$ and $\mathbf {y}$ as

$$\begin{aligned} \mathbf {c}_1=(1+\epsilon )^{-1}(\mathbf {x}+\mathbf {y}), \quad \mathbf {c}_2=(1+\epsilon )^{-1}(\mathbf {y}-\epsilon \mathbf {x}). \end{aligned}$$

(2.134)

The integral $I(\epsilon )$ can now be easily computed with the result

$$\begin{aligned} I(\epsilon )= & {} (1+\epsilon )^{-d}S_d^2\int _{0}^\infty {\mathrm {d}}x\; x^{d+2}{\mathrm {e}}^{-\epsilon x^2/(1+\epsilon )}\int _{0}^\infty {\mathrm {d}}y\; y^{d-1} {\mathrm {e}}^{- y^2/(1+\epsilon )}\nonumber \\= & {} \pi ^{d} \frac{\varGamma \left( \frac{d+3}{2}\right) }{\varGamma \left( d/2\right) }\epsilon ^{-d/2}\left( \frac{1+\epsilon }{\epsilon }\right) ^{3/2}, \end{aligned}$$

(2.135)

where $S_d=2\pi ^{d/2}/\varGamma (d/2)$ is the surface area of a d-dimensional unit sphere and use has been made of the symmetry properties of the Gaussian integrals. Use of the result (2.135) in Eq. (2.131) yields

$$\begin{aligned} \mu _2=\frac{S_d}{2\sqrt{2\pi }}(1-\alpha ^2)\left( 1+\frac{3}{16}a_2\right) . \end{aligned}$$

(2.136)

This expression agrees with Eqs. (2.32) and (2.33) when $a_3=0$. The remaining collision integrals appearing in the monocomponent case can be determined by following similar mathematical steps.

The partial cooling rates $\zeta _i^*$ of a multicomponent granular mixture are defined by Eq. (2.62). The general property (1.172) leads to

$$\begin{aligned} \zeta _i^*= & {} -\frac{2}{d}\theta _i\sum _j x_j \chi _{ij} \left( \frac{\sigma _{ij}}{\overline{\sigma }}\right) ^{d-1}\mu _{ji}(1+\alpha _{ij}) \int {\mathrm {d}}\mathbf {c}_1 \int \ {\mathrm {d}}\mathbf{c}_{2}\int {\mathrm {d}}\widehat{\varvec{\sigma }} \varTheta (\widehat{{\varvec{\sigma }}}\cdot \mathbf{g}_{12}^*)(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*)^2 \nonumber \\&\times \varphi _i(c_1) \varphi _j(c_2)\left[ \mu _{ji}(1+\alpha _{ij})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*)-2 (\widehat{\varvec{\sigma }}\cdot \mathbf{c}_1)\right] , \end{aligned}$$

(2.137)

where use has been made of the relation

$$\begin{aligned} c_1'^{2}-c_1^2=\mu _{ji}(1+\alpha _{ij})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*) \left[ \mu _{ji}(1+\alpha _{ij})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*)-2 (\widehat{\varvec{\sigma }}\cdot \mathbf{c}_1)\right] . \end{aligned}$$

(2.138)

Integration in (2.137) over the unit vector $\widehat{\varvec{\sigma }}$ can be carried out by employing the result (1.177) and

$$\begin{aligned} \int {\mathrm {d}}\widehat{\varvec{\sigma }}\, \varTheta (\widehat{{\varvec{\sigma }}} \cdot \mathbf{g}_{12}^*)\, (\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*)^k \widehat{\varvec{\sigma }}=B_{k+1} g_{12}^{* (k-1)}{} \mathbf{g}_{12}^*, \quad k\ge 0. \end{aligned}$$

(2.139)

With these results, Eq. (2.137) becomes

$$\begin{aligned} \zeta _i^*= & {} -\frac{2}{d}\theta _iB_3\sum _j x_j \chi _{ij} \left( \frac{\sigma _{ij}}{\overline{\sigma }}\right) ^{d-1}\mu _{ji}(1+\alpha _{ij}) \int {\mathrm {d}}\mathbf {c}_1 \int \ {\mathrm {d}}\mathbf{c}_{2} \varphi _i(c_1) \varphi _j(c_2)g_{12}^*\nonumber \\&\times \left[ \mu _{ji}(1+\alpha _{ij})g_{12}^{*2}-2 (\mathbf {c}_1 \cdot \mathbf {g}_{12}^*)\right] . \end{aligned}$$

(2.140)

Expression (2.140) is still exact. By substituting the leading Sonine approximation (2.68) into Eq. (2.140) and neglecting nonlinear terms in $a_2^{(i)}$ we obtain

$$\begin{aligned} \zeta _i^*=-\frac{2}{d}\theta _iB_3\sum _j x_j \chi _{ij} \left( \frac{\sigma _{ij}}{\overline{\sigma }}\right) ^{d-1}\mu _{ji}(1+\alpha _{ij})(\theta _i \theta _j)^{d/2} \left( 1+\var** _i+\var** _j\right) I_\zeta (\theta _i,\theta _j), \end{aligned}$$

(2.141)

where $B_3=\displaystyle {\frac{\pi ^{(d-1)/2}}{\varGamma \left( \frac{d+3}{2}\right) }}$,

$$\begin{aligned} I_\zeta (\theta _i,\theta _j) = \pi ^{-d}\int {\mathrm {d}}\mathbf {c}_1 \int \ {\mathrm {d}}\mathbf{c}_{2} \, {\mathrm {e}}^{-(\theta _{i}c_{1}^{2}+\theta _{j}c_{2}^{2})} g_{12}^*\left[ \mu _{ji}(1+\alpha _{ij})g_{12}^{*2}-2 (\mathbf {c}_1 \cdot \mathbf {g}_{12}^*)\right] , \end{aligned}$$

(2.142)

and the following operator has been introduced

$$\begin{aligned} \var** _i= \frac{a_2^{(i)}}{2}\left[ \theta _i^2\frac{\partial ^2}{\partial \theta _i^2}+(d+2)\theta _i\frac{\partial }{\partial \theta _i}+\frac{d(d+2)}{4}\right] . \end{aligned}$$

(2.143)

The integral (2.142) can be performed by the change of variables $(\mathbf {c}_1,\mathbf {c}_2) \rightarrow (\mathbf {x}, \mathbf {y})$ where

$$\begin{aligned} \mathbf {x}=\mathbf {c}_{1}-\mathbf {c}_{2}, \quad \mathbf {y}=\theta _{i} \mathbf {c}_{1}+\theta _{j}\mathbf {c}_{2} \end{aligned}$$

(2.144)

with the Jacobian $\left( \theta _{i}+\theta _{j}\right) ^{-d}$. With this change the integral $I_{\zeta }(\theta _i,\theta _j)$ can be easily computed and the result is

$$\begin{aligned} I_{\zeta }(\theta _i,\theta _j)=&\; -2\frac{\varGamma \left( \frac{d+3}{2}\right) }{\varGamma \left( \frac{d}{2}\right) }\left( \theta _i\theta _j\right) ^{-d/2}\theta _i^{-3/2} (1+\theta _{ij})^{1/2}\nonumber \\&\left[ 1-\frac{1}{2}\mu _{ji}(1+\alpha _{ij})(1+\theta _{ij})\right] , \end{aligned}$$

(2.145)

where $\theta _{ij} = \theta _i/\theta _j=m_i T_j/m_j T_i$ is the ratio between the mean-square velocity of the particles of species j relative to that of the particles of species i. Use of (2.141) in (2.145) leads to the final expressions of $\zeta _i^{(0)}$ and $\zeta _{ij}^{(1)}$. Their explicit forms can be found in Ref. [70]. In particular, the leading term $\zeta _i^{(0)}$ is

$$\begin{aligned} \zeta _i^{(0)}=\frac{4\pi ^{\frac{d-1}{2}}}{d\varGamma \left( \frac{d}{2}\right) }\theta _i^{-1/2}\sum _j x_j \chi _{ij}&\left( \frac{\sigma _{ij}}{\overline{\sigma }}\right) ^{d-1}\mu _{ji}(1+\alpha _{ij})(1+\theta _{ij})^{1/2} \nonumber \\&\times \left[ 1-\frac{1}{2}\mu _{ji}(1+\alpha _{ij})(1+\theta _{ij}) \right] . \end{aligned}$$

(2.146)

The fourth-degree collisional moments $\varLambda _i$ can be evaluated by following similar mathematical steps to those made for $\zeta _i^*$. First, by applying the scattering rule $\mathbf {c}_1'=\mathbf {c}_1-\mu _{ji}(1+\alpha _{ij})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*)\widehat{\varvec{\sigma }}$, we have the identity

$$\begin{aligned} c_1'^4-c_1^4= & {} 2\mu _{ji}^2(1+\alpha _{ij})^2(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*)^2 \left[ 2(\widehat{\varvec{\sigma }}\cdot \mathbf{c}_{1})^2+c_1^2+\frac{\mu _{ji}^2}{2} (1+\alpha _{ij})^2(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*)^2\right] \nonumber \\&-4\mu _{ji}(1+\alpha _{ij})(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*) (\widehat{\varvec{\sigma }}\cdot \mathbf{c}_{1})\left[ c_1^2+ \mu _{ji}^2(1+\alpha _{ij})^2(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*)^2\right] . \end{aligned}$$

(2.147)

Next, to perform the angular integrations on the right side of (2.67), we need to use the result (2.139) and

$$\begin{aligned} \int {\mathrm {d}}\widehat{\varvec{\sigma }}\,(\widehat{\varvec{\sigma }}\cdot \mathbf{g}_{12}^*)^k \widehat{\varvec{\sigma }}\widehat{\varvec{\sigma }} =\frac{B_{k}}{k+d} g_{12}^{*(k-2)}\left( k\mathbf{g}_{12}^*\mathbf{g}_{12}^*+g_{12}^{*2} \mathsf {I}\right) . \end{aligned}$$

(2.148)

These results allow us to carry out all the angular integrations and write $\varLambda _i$ as

$$\begin{aligned} \varLambda _i=\frac{B_3}{d+3} \sum _{j} x_j\chi _{ij} \left( \frac{\sigma _{ij}}{\overline{\sigma }}\right) ^{d-1} \mu _{ji}(1+\alpha _{ij}) \left( \theta _i\theta _j\right) ^{d/2}\left( 1+\var** _i+\var** _j\right) I_{\varLambda }(\theta _i,\theta _j), \end{aligned}$$

(2.149)

where the operator $\var** _i$ is defined by Eq. (2.143) and $I_{\varLambda }(\theta _i,\theta _j)$ is

$$\begin{aligned} I_{\varLambda }(\theta _i,\theta _j)= & {} \pi ^{-d} \int \,{\mathrm {d}}\mathbf {c}_{1}\,\int \,{\mathrm {d}}\mathbf{c}_{2}\,\, {\mathrm {e}}^{-(\theta _{i}c_{1}^{2}+\theta _{j}c_{2}^{2})}g_{12}^*\nonumber \\&\times \Bigg \{4 \mu _{ji}(1+\alpha _{ij})\left[ 3(\mathbf{c}_1\cdot \mathbf{g}_{12}^*)^2+\frac{d+5}{2}g_{12}^{*2}c_1^2+ \mu _{ji}^2(1+\alpha _{ij})^2g_{12}^{*4}\right] \nonumber \\&-4 (\mathbf{c}_1\cdot \mathbf{g}_{12}^*)\left[ (d+3)c_1^2+4 \mu _{ji}^2(1+\alpha _{ij})^2g_{12}^{*2}\right] \Bigg \}. \end{aligned}$$

(2.150)

The integral $I_{\varLambda }(\theta _i,\theta _j)$ can be performed by the change of variables (2.144). With this change, the integrations can be performed quite efficiently by using a computer package of symbolic calculation. The result is

$$\begin{aligned} I_{\varLambda }(\theta _i,\theta _j)= & {} (d+3) \frac{\varGamma \left( \frac{d+3}{2}\right) }{\varGamma \left( \frac{d}{2}\right) }\left( \theta _i\theta _j\right) ^{-d/2}\theta _i^{-5/2} (1+\theta _{ij})^{-1/2} \nonumber \\&\times \Bigg \{ -2\left[ d+3+(d+2)\theta _{ij}\right] +\mu _{ji}\left( 1+\alpha _{ij}\right) \left( 1+\theta _{ij} \right) \nonumber \\&\times \left( 11+ d+\frac{d^2+5d+6}{d+3} \theta _{ij} \right) -8\mu _{ji}^{2}\left( 1+\alpha _{ij}\right) ^{2}\left( 1+\theta _{ij} \right) ^{2} \nonumber \\&\,+2\mu _{ji}^{3}\left( 1+\alpha _{ij}\right) ^{3}\left( 1+\theta _{ij} \right) ^{3}\Bigg \}. \end{aligned}$$

(2.151)

Use of (2.151) in (2.149) leads to the explicit forms of $\varLambda _i^{(0)}$ and $\varLambda _{ij}^{(1)}$ [70].