Inelastic Maxwell Models (IMM)

Updated 14 July 2026

Inelastic Maxwell Models are simplified kinetic models for granular gases that use a constant collision frequency to enable analytical closure.
They provide exact evaluations of collisional moments and transport coefficients, facilitating precise studies of hydrodynamic and non-Newtonian flow behaviors.
The models identify critical inelastic thresholds where standard hydrodynamics break down, offering actionable insights for granular flow research.

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 $d$ (Brey et al., 2010). 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 (Garzó et al., 2010, Khalil et al., 2014, Kremer et al., 2022).

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 (Khalil et al., 2014). 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 $0\le \alpha\le 1$ (Khalil et al., 2014).

Relative to inelastic hard spheres (IHS), IMM replace the standard inelastic Boltzmann collision rate proportional to $d$ 0 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 (Brey et al., 2010, Garzó et al., 2010, Kubicki et al., 2014).

The hydrodynamic fields are the number density $d$ 1, flow velocity $d$ 2, and granular temperature $d$ 3, defined as moments of $d$ 4. The pressure tensor and heat flux are

$d$ 5

with $d$ 6. IMM conserve mass and momentum but dissipate energy through a cooling rate $d$ 7 that is exactly proportional to the collision frequency. In the normalization used in generalized IMM,

$d$ 8

while in other normalizations the proportionality constant is written differently because $d$ 9 is defined differently (Khalil et al., 2014, Garzó et al., 2010).

2. Homogeneous cooling state and spectral structure

IMM admit a similarity solution, the homogeneous cooling state (HCS),

$\partial_t f(\mathbf{r},\mathbf{v},t)+\mathbf{v}\cdot\nabla f(\mathbf{r},\mathbf{v},t)=J[\mathbf{v}|f,f],$ 0

where $\partial_t f(\mathbf{r},\mathbf{v},t)+\mathbf{v}\cdot\nabla f(\mathbf{r},\mathbf{v},t)=J[\mathbf{v}|f,f],$ 1 is isotropic and $\partial_t f(\mathbf{r},\mathbf{v},t)+\mathbf{v}\cdot\nabla f(\mathbf{r},\mathbf{v},t)=J[\mathbf{v}|f,f],$ 2 obeys $\partial_t f(\mathbf{r},\mathbf{v},t)+\mathbf{v}\cdot\nabla f(\mathbf{r},\mathbf{v},t)=J[\mathbf{v}|f,f],$ 3 (Brey et al., 2010). Thus the HCS temperature decays exponentially in scaled time $\partial_t f(\mathbf{r},\mathbf{v},t)+\mathbf{v}\cdot\nabla f(\mathbf{r},\mathbf{v},t)=J[\mathbf{v}|f,f],$ 4, but algebraically in real time due to $\partial_t f(\mathbf{r},\mathbf{v},t)+\mathbf{v}\cdot\nabla f(\mathbf{r},\mathbf{v},t)=J[\mathbf{v}|f,f],$ 5. In the formulation of Garzó and Santos, Haff’s law holds: $\partial_t f(\mathbf{r},\mathbf{v},t)+\mathbf{v}\cdot\nabla f(\mathbf{r},\mathbf{v},t)=J[\mathbf{v}|f,f],$ 6 so IMM do not produce an exponential decay in HCS in real time (Garzó et al., 2010).

The HCS also displays non-Gaussian high-velocity structure. In $\partial_t f(\mathbf{r},\mathbf{v},t)+\mathbf{v}\cdot\nabla f(\mathbf{r},\mathbf{v},t)=J[\mathbf{v}|f,f],$ 7, the scaled HCS distribution has a heavy tail $\partial_t f(\mathbf{r},\mathbf{v},t)+\mathbf{v}\cdot\nabla f(\mathbf{r},\mathbf{v},t)=J[\mathbf{v}|f,f],$ 8, and the exact fourth cumulant is

$\partial_t f(\mathbf{r},\mathbf{v},t)+\mathbf{v}\cdot\nabla f(\mathbf{r},\mathbf{v},t)=J[\mathbf{v}|f,f],$ 9

which is positive and increasing with inelasticity (Garzó et al., 2010). In one dimension, the HCS distribution is exactly

whose moments of order $J[\mathbf{v}_1|f,f]=\frac{(d+2)\nu_0}{2n\,\Omega_d}\int d\mathbf{v}_2\int d\boldsymbol{\sigma} \left[\alpha^{-1}f(\mathbf{r},\mathbf{v}_1',t)f(\mathbf{r},\mathbf{v}_2',t)-f(\mathbf{r},\mathbf{v}_1,t)f(\mathbf{r},\mathbf{v}_2,t)\right],$ 1 diverge (Garzó et al., 2010).

A distinctive feature of IMM is the explicit spectral analysis of the linearized homogeneous Boltzmann equation about the HCS. In scaled variables,

corresponding to density, flow, and temperature perturbations (Brey et al., 2010). The right hydrodynamic eigenfunctions are

while the left eigenfunctions are

with biorthogonality $J[\mathbf{v}_1|f,f]=\frac{(d+2)\nu_0}{2n\,\Omega_d}\int d\mathbf{v}_2\int d\boldsymbol{\sigma} \left[\alpha^{-1}f(\mathbf{r},\mathbf{v}_1',t)f(\mathbf{r},\mathbf{v}_2',t)-f(\mathbf{r},\mathbf{v}_1,t)f(\mathbf{r},\mathbf{v}_2,t)\right],$ 7 (Brey et al., 2010).

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

The Chapman–Enskog method for IMM starts from a normal solution expanded in gradients,

with $J[\mathbf{v}_1|f,f]=\frac{(d+2)\nu_0}{2n\,\Omega_d}\int d\mathbf{v}_2\int d\boldsymbol{\sigma} \left[\alpha^{-1}f(\mathbf{r},\mathbf{v}_1',t)f(\mathbf{r},\mathbf{v}_2',t)-f(\mathbf{r},\mathbf{v}_1,t)f(\mathbf{r},\mathbf{v}_2,t)\right],$ 9 the local HCS (Khalil et al., 2014). Because exact collisional moments are available, the Navier–Stokes constitutive relations can be derived without Sonine expansions: $n$ 0 For generalized IMM,

$n$ 1

with

$n$ 2

and

$n$ 3

(Khalil et al., 2014). Garzó and Santos also gave explicit IMM expressions for $n$ 4, $n$ 5, $n$ 6, the modified thermal conductivity $n$ 7, and the self-diffusion coefficient $n$ 8 in the $n$ 9 normalization that mimics IHS (Garzó et al., 2010).

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 $\Omega_d=2\pi^{d/2}/\Gamma(d/2)$ 0, $\Omega_d=2\pi^{d/2}/\Gamma(d/2)$ 1, and $\Omega_d=2\pi^{d/2}/\Gamma(d/2)$ 2. 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

$\Omega_d=2\pi^{d/2}/\Gamma(d/2)$ 3

and the exact Burnett structure can be used to estimate Burnett coefficients for IHS by replacing IMM inputs with IHS Sonine approximations (Khalil et al., 2014).

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

$\Omega_d=2\pi^{d/2}/\Gamma(d/2)$ 4

$\Omega_d=2\pi^{d/2}/\Gamma(d/2)$ 5

Hydrodynamics at long times requires all kinetic modes to decay faster than the slowest hydrodynamic mode $\Omega_d=2\pi^{d/2}/\Gamma(d/2)$ 6. Solving $\Omega_d=2\pi^{d/2}/\Gamma(d/2)$ 7 yields

$\Omega_d=2\pi^{d/2}/\Gamma(d/2)$ 8

so $\Omega_d=2\pi^{d/2}/\Gamma(d/2)$ 9 in $\nu_0\propto nT^\gamma$ 0 and $\nu_0\propto nT^\gamma$ 1 in $\nu_0\propto nT^\gamma$ 2. For $\nu_0\propto nT^\gamma$ 3, 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 (Brey et al., 2010). Near and below $\nu_0\propto nT^\gamma$ 4, the formally computed Chapman–Enskog heat conductivity diverges, and the same threshold appears in the Navier–Stokes and Burnett heat-flux sector (Brey et al., 2010, Khalil et al., 2014).

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

Uniform shear flow (USF) is characterized by constant $\nu_0\propto nT^\gamma$ 5, uniform $\nu_0\propto nT^\gamma$ 6, and linear flow $\nu_0\propto nT^\gamma$ 7. In the local Lagrangian frame, the distribution becomes spatially uniform and obeys

$\nu_0\propto nT^\gamma$ 8

The energy balance is

$\nu_0\propto nT^\gamma$ 9

so viscous heating competes with inelastic cooling (Garzó et al., 2010). For dry IMM, Garzó and Santos obtained exact steady USF pressure tensor elements,

as well as the exact generalized viscosity and viscometric function,

As $\mathbf{v}_1'=\mathbf{v}_1-\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},\qquad \mathbf{v}_2'=\mathbf{v}_2+\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},$ 3 decreases, $\mathbf{v}_1'=\mathbf{v}_1-\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},\qquad \mathbf{v}_2'=\mathbf{v}_2+\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},$ 4 and $\mathbf{v}_1'=\mathbf{v}_1-\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},\qquad \mathbf{v}_2'=\mathbf{v}_2+\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},$ 5 decrease monotonically, reflecting shear thinning and normal stress differences far from the Navier–Stokes regime (Garzó et al., 2010).

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, $\mathbf{v}_1'=\mathbf{v}_1-\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},\qquad \mathbf{v}_2'=\mathbf{v}_2+\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},$ 6, $\mathbf{v}_1'=\mathbf{v}_1-\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},\qquad \mathbf{v}_2'=\mathbf{v}_2+\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},$ 7, and $\mathbf{v}_1'=\mathbf{v}_1-\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},\qquad \mathbf{v}_2'=\mathbf{v}_2+\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},$ 8, with the hallmark property $\mathbf{v}_1'=\mathbf{v}_1-\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},\qquad \mathbf{v}_2'=\mathbf{v}_2+\frac{1}{2}(1+\alpha^{-1})(\mathbf{g}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma},$ 9. All second-order moments coincide with the steady USF values, while the third-order moments define two generalized heat-flux coefficients $\mathbf{g}=\mathbf{v}_1-\mathbf{v}_2$ 0 and $\mathbf{g}=\mathbf{v}_1-\mathbf{v}_2$ 1. In $\mathbf{g}=\mathbf{v}_1-\mathbf{v}_2$ 2, both diverge at the same $\mathbf{g}=\mathbf{v}_1-\mathbf{v}_2$ 3 at which fourth-order USF moments diverge (Garzó et al., 2010).

Small spatial perturbations of USF require tensorial transport coefficients. The generalized constitutive equations are

$\mathbf{g}=\mathbf{v}_1-\mathbf{v}_2$ 4

and, for IMM, the corresponding equations can be solved exactly as functions of shear rate, restitution coefficients, and mixture parameters (Garzó et al., 2010, Garzó et al., 2015).

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,

$\mathbf{g}=\mathbf{v}_1-\mathbf{v}_2$ 5

and the steady energy balance becomes

$\mathbf{g}=\mathbf{v}_1-\mathbf{v}_2$ 6

The exact IMM predictions for the rheological properties show an $\mathbf{g}=\mathbf{v}_1-\mathbf{v}_2$ 7 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 $\mathbf{g}=\mathbf{v}_1-\mathbf{v}_2$ 8 and $\mathbf{g}=\mathbf{v}_1-\mathbf{v}_2$ 9, and are unphysical in between (González et al., 2018).

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 $0\le \alpha\le 1$ 0. 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 (Garzó et al., 2011, Garzó et al., 2015). In uniform shear flow, the long-time temperature evolves as

$0\le \alpha\le 1$ 1

with $0\le \alpha\le 1$ 2 determined by a sixth-degree polynomial arising from the coupled moment equations (Garzó et al., 2011).

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 $0\le \alpha\le 1$ 3 and $0\le \alpha\le 1$ 4. If $0\le \alpha\le 1$ 5, the phase is disordered: $0\le \alpha\le 1$ 6 and $0\le \alpha\le 1$ 7 remains finite. If $0\le \alpha\le 1$ 8, the phase is ordered: $0\le \alpha\le 1$ 9 remains finite while $d$ 00. 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 (Garzó et al., 2011).

The anisotropy induced by shear requires tensorial diffusion coefficients instead of scalar ones. Around USF, the tracer mass flux takes the form

$d$ 01

and the tensors $d$ 02, $d$ 03, and $d$ 04 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 $d$ 05, the coefficients of restitution $d$ 06, and the parameters of the mixture (Garzó et al., 2016, Garzó et al., 2015). In the disordered tracer phase, $d$ 07; in the ordered phase, they become finite and inherit the sensitivity of the order–disorder transition (Garzó et al., 2016).

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

$d$ 08

with exact expressions for $d$ 09, $d$ 10, $d$ 11, and $d$ 12. 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 (Khalil et al., 2018). 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 $d$ 13, tangential restitution $d$ 14, and reduced moment of inertia $d$ 15, and the Maxwell kernel again makes collisional moments exactly computable (Kremer et al., 2022). 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 (Kremer et al., 2022).

A recent exact USF solution for inelastic and rough Maxwell particles expresses the rheology in terms of two effective parameters $d$ 16 and $d$ 17. The rotational-to-translational temperature ratio $d$ 18 and the proportionality $d$ 19 are independent of $d$ 20 and determined solely by roughness and moment of inertia. The steady reduced stresses are

$d$ 21

with

$d$ 22

In the appropriate limits, the results reduce to the smooth IMM and to the Pidduck gas in the elastic perfectly rough case (Santos et al., 26 Feb 2026).

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 $d$ 23; for Ornstein–Uhlenbeck driving, steady states exist whenever $d$ 24. The stationary velocity distributions have exact large-velocity tails: exponential for $d$ 25 and Gaussian for $d$ 26 (Prasad et al., 2014). 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 $d$ 27, 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 (Prasad et al., 2017). 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 (Prasad et al., 2016).

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 $d$ 28 over $d$ 29. IMM exhibit algebraic high-velocity tails in the HCS, possible divergence of heat-flux and higher-order moments, $d$ 30 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 (Brey et al., 2010, Garzó et al., 2010, González et al., 2018). 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.