---
title: Inelastic Maxwell Models (IMM)
url: https://www.emergentmind.com/topics/inelastic-maxwell-models-imm
type: topic
---

# Inelastic Maxwell Models (IMM)

Inelastic Maxwell Models (IMM) are simplified kinetic models for dilute granular gases that retain the essential physics of inelastic collisions while making the collision operator analytically tractable. Their defining simplifications are a constant, or effectively constant, collision frequency, a velocity-independent collision kernel, and an angle-independent cross section; all dependence on inelasticity enters through the coefficient of normal restitution \(\alpha\) and the dimensionality \(d\) [1004.4742]. This tractability makes possible exact evaluation of collisional moments, explicit hydrodynamic constitutive equations, and closed analyses of homogeneous cooling, non-Newtonian shear states, tracer transport, driven steady states, and rough-particle generalizations [1004.4453] [1401.7933] [2203.10119].

## 1. Definition and kinetic framework

In IMM, the one-particle distribution function obeys an inelastic Boltzmann equation in which the collision rate is independent of the magnitude of the relative velocity. A general form is
\[
\partial_t f(\mathbf{r},\mathbf{v},t)+\mathbf{v}\cdot\nabla f(\mathbf{r},\mathbf{v},t)=J[\mathbf{v}|f,f],
\]
with collision operator
\[
J[\mathbf{v}_1|f,f]=\frac{(d+2)\nu_0}{2n\,\Omega_d}\int d\mathbf{v}_2\int d\boldsymbol{\sigma}
\left[\alpha^{-1}f(\mathbf{r},\mathbf{v}_1',t)f(\mathbf{r},\mathbf{v}_2',t)-f(\mathbf{r},\mathbf{v}_1,t)f(\mathbf{r},\mathbf{v}_2,t)\right],
\]
where \(n\) is the number density, \(\Omega_d=2\pi^{d/2}/\Gamma(d/2)\), and \(\nu_0\propto nT^\gamma\) in generalized IMM [1401.7933]. For smooth particles, the restituting velocities are
\[
\mathbf{v}_1'=\mathbf{v}_1-\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},\qquad
\mathbf{v}_2'=\mathbf{v}_2+\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},
\]
with \(\mathbf{g}=\mathbf{v}_1-\mathbf{v}_2\) and \(0\le \alpha\le 1\) [1401.7933].

Relative to inelastic hard spheres (IHS), IMM replace the standard inelastic Boltzmann collision rate proportional to \(|\mathbf{g}\cdot\widehat{\boldsymbol{\sigma}}|\) by an effective average collision rate independent of velocities and angles. This replacement is the source of the model’s analytic closure properties. A central advantage of IMM over IHS is that velocity moments of the Boltzmann collision operator can be exactly expressed in terms of lower-order moments without needing the explicit form of the velocity distribution function. This allows exact Navier–Stokes and Burnett transport coefficients, exact second-degree collisional moments for mixtures, and exact nonlinear rheology in several shear-driven states [1004.4742] [1004.4453] [1411.3245].

The hydrodynamic fields are the number density \(n\), flow velocity \(\mathbf{u}\), and granular temperature \(T\), defined as moments of \(f\). The pressure tensor and heat flux are
\[
P_{ij}=m\int d\mathbf{v}\,V_iV_j f,\qquad
\mathbf{q}=\frac{m}{2}\int d\mathbf{v}\,V^2\mathbf{V}f,
\]
with \(\mathbf{V}=\mathbf{v}-\mathbf{u}\). IMM conserve mass and momentum but dissipate energy through a cooling rate \(\zeta\) that is exactly proportional to the collision frequency. In the normalization used in generalized IMM,
\[
\zeta=\frac{d+2}{4d}(1-\alpha^2)\nu_0,\qquad \zeta^*=\frac{\zeta}{\nu_0}=\frac{d+2}{4d}(1-\alpha^2),
\]
while in other normalizations the proportionality constant is written differently because \(\nu\) is defined differently [1401.7933] [1004.4453].

## 2. Homogeneous cooling state and spectral structure

IMM admit a similarity solution, the homogeneous cooling state (HCS),
\[
f_{\mathrm{HCS}}(\mathbf{v},t)=n_H\,v_0^{-d}(t)\,\chi(\mathbf{c}),\qquad
v_0(t)=\left(\frac{2T_H(t)}{m}\right)^{1/2},\qquad
\mathbf{c}=\frac{\mathbf{v}}{v_0(t)},
\]
where \(\chi(\mathbf{c})\) is isotropic and \(T_H\) obeys \(\partial_tT=-\zeta T\) [1004.4742]. Thus the HCS temperature decays exponentially in scaled time \(d\tau=\nu_H(t)\,dt\), but algebraically in real time due to \(\nu(t)\propto T^{1/2}\). In the formulation of Garzó and Santos, Haff’s law holds:
\[
T(t)=\frac{T(0)}{\left[1+\zeta(0)t/2\right]^2},
\]
so IMM do not produce an exponential decay in HCS in real time [1004.4453].

The HCS also displays non-Gaussian high-velocity structure. In \(d\ge 2\), the scaled HCS distribution has a heavy tail \(\phi_h(c)\sim c^{-(d+s(\alpha))}\), and the exact fourth cumulant is
\[
a_2=\frac{6(1-\alpha)^2}{4d-7+3\alpha(2-\alpha)},
\]
which is positive and increasing with inelasticity [1004.4453]. In one dimension, the HCS distribution is exactly
\[
\phi_h(c)=\frac{2\sqrt{2}}{\pi}\frac{1}{(1+2c^2)^2},
\]
whose moments of order \(>2\) diverge [1004.4453].

A distinctive feature of IMM is the explicit spectral analysis of the linearized homogeneous Boltzmann equation about the HCS. In scaled variables,
\[
\partial_\tau\,\delta\chi(\mathbf{c},\tau)=\Lambda(\mathbf{c})\,\delta\chi(\mathbf{c},\tau),
\]
with \(\Lambda\) non-Hermitian. The hydrodynamic eigenvalues are
\[
\lambda_1=0,\qquad \lambda_2=\frac{\widetilde{\zeta}}{2},\qquad \lambda_3=-\frac{\widetilde{\zeta}}{2},
\]
corresponding to density, flow, and temperature perturbations [1004.4742]. The right hydrodynamic eigenfunctions are
\[
\xi_1=(d+1)\chi+\mathbf{c}\cdot\frac{\partial\chi}{\partial\mathbf{c}},\qquad
\boldsymbol{\xi}_2=-\frac{\partial\chi}{\partial\mathbf{c}},\qquad
\xi_3=-\frac{\partial}{\partial\mathbf{c}}\cdot\big[\mathbf{c}\chi(\mathbf{c})\big],
\]
while the left eigenfunctions are
\[
\overline{\xi}_1=\chi,\qquad
\overline{\boldsymbol{\xi}}_2=\mathbf{c}\chi,\qquad
\overline{\xi}_3=\left(\frac{c^2}{d}+\frac{1}{2}\right)\chi,
\]
with biorthogonality \(\langle\overline{\xi}_i|\xi_j\rangle=\delta_{ij}\) [1004.4742].

## 3. Chapman–Enskog hydrodynamics, Burnett order, and breakdown

The Chapman–Enskog method for IMM starts from a normal solution expanded in gradients,
\[
f=f^{(0)}+\epsilon f^{(1)}+\epsilon^2 f^{(2)}+\cdots,\qquad
\partial_t=\partial_t^{(0)}+\epsilon\partial_t^{(1)}+\epsilon^2\partial_t^{(2)}+\cdots,
\]
with \(f^{(0)}\) the local HCS [1401.7933]. Because exact collisional moments are available, the Navier–Stokes constitutive relations can be derived without Sonine expansions:
\[
P_{ij}^{(1)}=-\eta\left(\nabla_i u_j+\nabla_j u_i-\frac{2}{d}\delta_{ij}\nabla\cdot\mathbf{u}\right),\qquad
\mathbf{q}^{(1)}=-\mu\,\nabla n-\kappa\,\nabla T.
\]
For generalized IMM,
\[
\eta^*=\frac{1}{\nu_{0|2}^*-(1-\gamma)\zeta^*},\qquad
\kappa^*=\frac{d-1}{d}\frac{1+2c}{\nu_{2|1}^*-2\zeta^*},\qquad
\mu^*=\frac{\kappa^*}{1+2c}\,\frac{\zeta^*+\nu_{2|1}^*c}{\nu_{2|1}^*-(2-\gamma)\zeta^*},
\]
with
\[
\nu_{0|2}^*=\frac{(1+\alpha)(d+1-\alpha)}{2d},\qquad
\nu_{2|1}^*=\frac{(1+\alpha)\big[5d+4-\alpha(d+8)\big]}{8d},
\]
and
\[
c=\frac{6(1-\alpha)^2}{4d-7+3\alpha(2-\alpha)}
\]
[1401.7933]. Garzó and Santos also gave explicit IMM expressions for \(\eta\), \(\kappa\), \(\mu\), the modified thermal conductivity \(\kappa'\), and the self-diffusion coefficient \(D\) in the \(q=1/2\) normalization that mimics IHS [1004.4453].

Khalil, Garzó, and Santos extended the IMM hydrodynamic description to Burnett order. The pressure tensor and heat flux were obtained to second order in the spatial gradients with explicit expressions for all Burnett transport coefficients as functions of \(\gamma\), \(\alpha\), and \(d\). Inelasticity breaks elastic degeneracies: coefficients that are related in a simple way in the elastic limit become decoupled in the inelastic case. A compact invariant relation is
\[
\frac{a_6}{a_7}=1-\frac{2(2-\gamma)}{d},
\]
and the exact Burnett structure can be used to estimate Burnett coefficients for IHS by replacing IMM inputs with IHS Sonine approximations [1401.7933].

The most distinctive limitation of IMM hydrodynamics is the explicit breakdown of time-scale separation. The left kinetic eigenfunctions associated with the pressure tensor and heat flux are
\[
\overline{\xi}_4=c_xc_y\,\chi,\qquad \overline{\lambda}_4=-\frac{(1+\alpha)^2}{4},
\]
\[
\overline{\xi}_5=\left(c^2-\frac{d+2}{2}\right)c_x\,\chi,\qquad
\overline{\lambda}_5=-\frac{(d-1)(1+\alpha)^2}{4d}.
\]
Hydrodynamics at long times requires all kinetic modes to decay faster than the slowest hydrodynamic mode \(\lambda_3=-\widetilde{\zeta}/2\). Solving \(|\lambda_5|=\widetilde{\zeta}/2\) yields
\[
\alpha_c=\frac{4-d}{3d},
\]
so \(\alpha_c=1/3\) in \(d=2\) and \(\alpha_c=1/9\) in \(d=3\). For \(\alpha\le \alpha_c\), the heat-flux kinetic mode decays more slowly than the slowest hydrodynamic mode, the hydrodynamic spectrum is not isolated, and there is no closed hydrodynamic description [1004.4742]. Near and below \(\alpha_c\), the formally computed Chapman–Enskog heat conductivity diverges, and the same threshold appears in the Navier–Stokes and Burnett heat-flux sector [1004.4742] [1401.7933].

## 4. Uniform shear flow, Couette flow, and non-Newtonian rheology

Uniform shear flow (USF) is characterized by constant \(n\), uniform \(T\), and linear flow \(\mathbf{u}=ay\,\widehat{\mathbf{x}}\). In the local Lagrangian frame, the distribution becomes spatially uniform and obeys
\[
\partial_t f(\mathbf{V})-aV_y\frac{\partial f}{\partial V_x}=J[f,f].
\]
The energy balance is
\[
\partial_t T=-\frac{2}{dn}P_{xy}a-\zeta T,
\]
so viscous heating competes with inelastic cooling [1004.4453]. For dry IMM, Garzó and Santos obtained exact steady USF pressure tensor elements,
\[
P_{yy}^*=\frac{d}{2}\frac{1+\alpha}{d+1-\alpha},\qquad
P_{xy}^*=-\frac{d\sqrt{(d+2)(1-\alpha^2)}}{2\sqrt{2}(d+1-\alpha)},
\]
\[
P_{xx}^*=\frac{d}{2}\frac{d+3-(d+1)\alpha}{d+1-\alpha},
\]
as well as the exact generalized viscosity and viscometric function,
\[
\eta(\alpha,d)=\eta_0\left[\frac{d}{d+1-\alpha}\right]^2,\qquad
\Psi(\alpha,d)=\Psi_0\,\frac{2}{1+\alpha}\left[\frac{d}{d+1-\alpha}\right]^3.
\]
As \(\alpha\) decreases, \(\eta/\eta_0\) and \(\Psi/\Psi_0\) decrease monotonically, reflecting shear thinning and normal stress differences far from the Navier–Stokes regime [1004.4453].

IMM also admit an exact solution for a special class of Couette flows characterized by a uniform heat flux, usually termed LTu flow. In this state, \(p=\mathrm{const}\), \(\partial_su_x=a=\mathrm{const}\), and \(\partial_sT=\mathrm{const}\), with the hallmark property \(\partial T/\partial u_x=\mathrm{const}\). All second-order moments coincide with the steady USF values, while the third-order moments define two generalized heat-flux coefficients \(\kappa\) and \(\Phi\). In \(d=3\), both diverge at the same \(\alpha_c\simeq 0.046\) at which fourth-order USF moments diverge [1004.4453].

Small spatial perturbations of USF require tensorial transport coefficients. The generalized constitutive equations are
\[
P_{ij}^{(1)}=-\eta_{ijk\ell}\nabla_\ell \delta u_k,\qquad
q_i^{(1)}=-\kappa_{ij}\nabla_j T-\mu_{ij}\nabla_j n,
\]
and, for IMM, the corresponding equations can be solved exactly as functions of shear rate, restitution coefficients, and mixture parameters [1004.4453] [1506.08677].

When interstitial gas effects are modeled by drag and stochastic forcing, IMM still provide exact second- and fourth-degree collisional moments. For granular suspensions under simple shear flow,
\[
-aV_y\frac{\partial f}{\partial V_x}-\gamma\frac{\partial}{\partial\mathbf{V}}\cdot(\mathbf{V}f)-\frac{\gamma T_{\rm ex}}{m}\frac{\partial^2 f}{\partial V^2}=J_{\mathrm{IMM}}[f,f],
\]
and the steady energy balance becomes
\[
-\frac{2}{dn}aP_{xy}-\zeta T+2\gamma(T_{\rm ex}-T)=0.
\]
The exact IMM predictions for the rheological properties show an \(S\) shape in a plane of stress–strain rate, corresponding to discontinuous shear thickening, and agree excellently with event-driven simulations for IHS. At the same time, fourth-degree moments can diverge at two critical temperatures \(\theta_c^{(1)}(\alpha)\) and \(\theta_c^{(2)}(\alpha)\), and are unphysical in between [1811.05859].

## 5. Mixtures, tracer dynamics, and driven states

For low-density granular binary mixtures, IMM replace the velocity-dependent hard-sphere collision rate by a velocity-independent one while retaining species masses, mole fractions, and restitution coefficients \(\alpha_{rs}\). This permits exact collisional closure for second-degree moments and hence exact pressure tensors, cooling rates, and transport coefficients in homogeneous shear and near-shear states [1112.2036] [1506.08677]. In uniform shear flow, the long-time temperature evolves as
\[
T(t)=T(0)e^{\lambda \nu_0 t},
\]
with \(\lambda\) determined by a sixth-degree polynomial arising from the coupled moment equations [1112.2036].

In the tracer limit, a non-equilibrium phase transition appears. For an impurity immersed in a sheared inelastic Maxwell gas, the characteristic polynomial factorizes into a host-gas cubic and an impurity cubic, with largest roots \(\lambda_2^{(0)}\) and \(\lambda_1^{(0)}\). If \(\lambda_2^{(0)}>\lambda_1^{(0)}\), the phase is disordered: \(E_1/E\to 0\) and \(T_1/T\) remains finite. If \(\lambda_1^{(0)}>\lambda_2^{(0)}\), the phase is ordered: \(E_1/E\) remains finite while \(T_1/T\to\infty\). Dissipation leads to new ordered phases, including a heavy-impurity ordered phase at small shear, while a light-impurity ordered phase appears above a critical shear rate [1112.2036].

The anisotropy induced by shear requires tensorial diffusion coefficients instead of scalar ones. Around USF, the tracer mass flux takes the form
\[
j_{1,i}^{(1)}=-\frac{m_1m_2n}{\rho}D_{ij}\frac{\partial x_1}{\partial r_j}
-\frac{\rho}{p}D_{p,ij}\frac{\partial p}{\partial r_j}
-\frac{\rho}{T}D_{T,ij}\frac{\partial T}{\partial r_j},
\]
and the tensors \(D_{ij}\), \(D_{p,ij}\), and \(D_{T,ij}\) are given in terms of the solutions of a set of coupled algebraic equations that can be exactly solved as functions of the shear rate \(a\), the coefficients of restitution \(\alpha_{sr}\), and the parameters of the mixture [1605.01529] [1506.08677]. In the disordered tracer phase, \(D_{p,ij}^*=D_{T,ij}^*=0\); in the ordered phase, they become finite and inherit the sensitivity of the order–disorder transition [1605.01529].

Driven homogeneous mixtures thermostatted by a drag force and a stochastic force provide another exact IMM setting. In that case, the Navier–Stokes mass-flux constitutive relation is
\[
\mathbf{j}_1^{(1)}=-\left(\frac{m_1m_2 n}{\rho}\right)D\,\nabla x_1
-\frac{\rho}{p}D_p\,\nabla p
-\frac{\rho}{T}D_T\,\nabla T
-D_U\,\Delta\mathbf{U},
\]
with exact expressions for \(D\), \(D_p\), \(D_T\), and \(D_U\). The temperature ratio in the homogeneous steady state shows excellent agreement with molecular dynamics simulations for driven IHS, even for strong inelasticity and/or disparity in masses and diameters [1809.06082]. This suggests that, for driven steady states, IMM can reproduce key non-equipartition and mass-transport trends of IHS while preserving exact moment closure.

## 6. Rough particles, one-dimensional driven IMM, and comparison with IHS

The IMM framework has been generalized from smooth particles to inelastic and rough Maxwell particles. In the inelastic rough Maxwell model (IRMM), particles carry both translational and rotational velocities, collisions are characterized by normal restitution \(\alpha\), tangential restitution \(\beta\), and reduced moment of inertia \(\kappa\), and the Maxwell kernel again makes collisional moments exactly computable [2203.10119]. The tractability of the proposed model is illustrated by the exact evaluation of the collisional moments of first and second degree, and the most relevant ones of third and fourth degree. These results are applied to the rotational-to-translational temperature ratio and the velocity cumulants in the HCS [2203.10119].

A recent exact USF solution for inelastic and rough Maxwell particles expresses the rheology in terms of two effective parameters \(\chi\) and \(\psi\). The rotational-to-translational temperature ratio \(\theta=T_r/T_t\) and the proportionality \(\Omega_{ij}^*=-\lambda \Pi_{ij}^*\) are independent of \(\alpha\) and determined solely by roughness and moment of inertia. The steady reduced stresses are
\[
\Pi_{xx}^*=2\chi/\psi,\qquad
\Pi_{yy}^*=\Pi_{zz}^*=-\chi/\psi,\qquad
\Pi_{xy}^*=-\sqrt{\frac{3}{2}\frac{\chi}{\psi}\left(1-\frac{\chi}{\psi}\right)},
\]
with
\[
\eta^*=\frac{1-\chi/\psi}{\psi},\qquad
\Psi_1=2\frac{1-\chi/\psi}{\psi^2}.
\]
In the appropriate limits, the results reduce to the smooth IMM and to the Pidduck gas in the elastic perfectly rough case [2602.22927].

IMM also admit exact driven one-dimensional realizations. In the continuous-time driven IMM without spatial structure, the hierarchy of kinetic equations for the velocity distributions does not close, yet the coupled evolution equations for the variance and two-particle velocity correlations do close exactly. For point-process driving, steady states exist whenever \(r_w\neq -1\); for Ornstein–Uhlenbeck driving, steady states exist whenever \(\Gamma\neq 0\). The stationary velocity distributions have exact large-velocity tails: exponential for \(r_w=1\) and Gaussian for \(|r_w|<1\) [1408.3964]. A complementary analysis of a driven one-component Maxwell gas shows that the steady-state velocity distribution is non-universal and depends strongly on the nature of driving; for dissipative wall driving \((|r_w|<1)\), the tail exponent equals that of the noise, while for diffusive driving the tail is universal only if the noise distribution decays faster than exponential [1701.03600]. In a lattice version with nearest-neighbor collisions and periodic boundary conditions, the equal-time spatial correlation decays exponentially with distance and the spatio-temporal correlation exhibits a ballistic front and a second-order discontinuity at moving transition points [1606.09561].

Comparison with IHS is central to the interpretation of IMM. The hydrodynamic eigenvalues in the homogeneous long-wavelength limit match those of IHS, but the kinetic spectrum differs significantly. In IHS, the pressure-tensor and heat-flux variables are not left eigenfunctions of the linearized inelastic Boltzmann operator, and the Navier–Stokes transport coefficients are regular functions of \(\alpha\) over \(0<\alpha\le 1\). IMM exhibit algebraic high-velocity tails in the HCS, possible divergence of heat-flux and higher-order moments, \(P_{yy}=P_{zz}\) identically in several shear problems, and even regimes in which hydrodynamics breaks down through loss of time-scale separation; such behavior is not expected for IHS [1004.4742] [1004.4453] [1811.05859]. This suggests that IMM are best understood as exactly solvable kinetic models that isolate mechanisms and parameter dependences, rather than as uniformly faithful surrogates of IHS across all moments and all inelasticities.

Source: https://www.emergentmind.com/topics/inelastic-maxwell-models-imm