In today’s blog, I will go into one of the issues in mathematical ecology mentioned in yesterday’s blog reporting on the MBI workshop on “Sustainability and Complex Systems.” The issue came up in the discussion sessions, where the question was asked how one could apply dimension-reduction techniques to individual-based models (IBMs) and derive more manageable descriptions of ecological systems.

The discussion at the workshop reminded me of my earlier interest in gas dynamics, where a similar issue arises: How to derive continuum models like the Navier-Stokes equations from the equations of motion of the individual molecules that make up the gas. This issue is one of the main topics of investigation in *kinetic theory*; it has a long history going back to Ludwig Boltzmann in the 1870s. While the initial discussions were mostly heuristic, mathematical research in the latter part of the 20th century has provided a rigorous framework for the various approximations, so today the theory is on a more or less solid foundation.

In kinetic theory, a gas is thought of as a collection of mutually interacting molecules, possibly moving under the influence of external forces. We assume for simplicity that the molecules are all of the same kind (a “simple gas”) and that there are no external forces acting on the molecules. Each molecule moves in physical space; at each instant $t$, its state is described by its position vector $x$ and its velocity vector $c$. Molecules interact, they may attract or repel each other, and as they interact their velocities change. The interactions are assumed to be local and instantaneous and derived from some potential (for example, the Lennard-Jones potential). If the interaction is elastic, then mass, momentum, and kinetic energy are preserved, so the velocities of two interacting molecules are determined uniquely in terms of their velocities before the interaction.

At the microscopic level, the state of a gas comprising $N$ molecules is described by an *$N$-particle distribution function* $f_N$ with values $f_N (\mathbf{x}_1, \dots, \mathbf{x}_N, \mathbf{c}_1, \dots, \mathbf{c}_N, t)$. This function evolves in a $6N$-dimensional space according to the *Liouville equation*,

$$

\frac{\partial f_N}{\partial t} + \sum_{i=1}^N (\nabla_{\mathbf{x}_i} f_N) \cdot \dot{\mathbf{x}}_i + \sum_{i=1}^N (\nabla_{\mathbf{c}_i} f_N) \cdot \mathbf{F}_i = 0 ,

\quad \mathbf{F}_i = {\ } – \sum_{j=1}^N \nabla_{\mathbf{x}_i} \Phi_{ij} .

$$

Note that the Liouville equation is linear in $f_N$.

By integration over part of the variables, the Liouville equation is transformed into a chain of $N$ equations where the first equation connects the evolution of one-particle distribution function with the two-particle distribution function, the second equation connects the two-particle distribution function with the three-particle distribution function, and generally the $s$th equation connects the $s$-particle distribution function $f_s$ with the $(s+1)$-particle distribution function $f_{s+1}$,

$$

\frac{\partial f_s}{\partial t} + \sum_{i=1}^s (\nabla_{\mathbf{x}_i} f_s) \cdot \dot{\mathbf{x}}_i + \sum_{i=1}^s (\nabla_{\mathbf{c}_i} f_s) \cdot \mathbf{F}_i = {\ } – \sum_{i=1}^s \nabla_{\mathbf{c}_i} \int (\nabla_{\mathbf{x}_i} \Phi_{i, s+1}) f_{s+1} \, d\mathbf{x}_{s+1} \, d\mathbf{c}_{s+1} .

$$

This is the so-called BBGKY hierarchy (named after its developers, Bogoliubov, Born, Green, Kirkwood and Yvon). which is a description of a gas at the *microscopic level*. The equations in the hierarchy define a linear operator in the space of chains of length $N$ of density functions.

From the BBGKY hierarchy one obtains a description at the *mesoscopic level* by taking the equation for the one-particle distribution function and employing a closure relation to express the two-particle distribution function as the product of two one-particle distribution functions. This is the (in)famous “Stosszahlansatz,” which leads to the *Boltzmann equation*—an integrodifferential equation for the one-particle distribution function $f_1$ (which we denote henceforth simply by $f$, with values $f(\mathbf{x}, \mathbf{c}, t)$) with a quadratic nonlinearity on the right-hand side,

$$

\frac{\partial f}{\partial t} + (\nabla_{\mathbf{x}} f) \cdot \dot{\mathbf{x}} + (\nabla_{\mathbf{c}} f) \mathbf{F} = \int \int (f’_1 f’_2 – f_1 f_2) k_{12} \, dk\, d\mathbf{x}_2 ,

$$

where $f’_1$ and $f’_2$ denote the values of $f$ at the velocity variables $\mathbf{c}’_1$ and $\mathbf{c}’_2$ before the interaction and $f_1$ and $f_2$ the same at the velocity variables $\mathbf{c}_1$ and $\mathbf{c}_2$ after the interaction. The kernel $k_{12}$ represents the change in direction of the relative velocity of the two molecules as a result of their interaction. The step from the BBGKY hierarchy to the Boltzmann equation introduces not only a nonlinearity, it also introduces *irreversibility*: the Boltzmann equation is time-irreversible (Boltzmann’s H-Theorem).

When mass, momentum and internal energy are preserved in a molecular interaction, a further reduction is possible. The macroscopic variables of the gas are the *mass density* $\rho$, the *hydrodynamic velocity* $\mathbf{v}$, and the *temperature* $T$ (which is a measure of the internal energy). They are the velocity moments of $f$ with respect to mass, momentum and internal energy. Multiplying both sides of the Boltzmann equation with the molecular mass and integrating over all velocities, we obtain the *continuity equation*,

$$

\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 .

$$

Multiplying both sides of the Boltzmann equation with the momentum vector and integrating over all velocities, we obtain the *equation of motion*,

$$

\frac{\partial (\rho \mathbf{v} )}{\partial t} + \nabla \cdot (\rho \mathbf{v} \mathbf{v}) – \nabla \cdot \mathbb{T} = 0 ,

$$

where $\mathbb{T}$ is the Cauchy stress tensor.

If the gas is in hydrostatic equilibrium, the Cauchy stress tensor is diagonal, the shear stresses are zero, and the normal stresses are all equal. The *hydrostatic pressure* $p$ is the negative of the normal stresses, so $\mathbb{T} = -p{\ } \mathbb{I}$. The equation of motion reduces to

$$

\frac{\partial (\rho \mathbf{v}}{\partial t} + \nabla \cdot (\rho \mathbf{v} \mathbf{v}) + \nabla p = 0 .

$$

This is is the Navier-Stokes equation of fluid dynamics, which describes the evolution of the gas at the *macroscopic level*.

The procedure outlined above to derive the Navier-Stokes equation from the Boltzmann equation is known as the *Chapman-Enskog procedure*. It is essentially an asymptotic analysis based on a two-time scale singular perturbation expansion, where the macroscopic variables evolve on the slow time scale and the one-particle distribution function on the fast time scale.

Thus, there exists a very systematic procedure to get from the microscopic level (the Liouville equation and the BBGKY hierarchy) to the mesoscopic level (the Boltzmann equation) and from there to the macroscopic level (the Navier-Stokes equation). Given that the IBMs are the analog in ecology of the Liouville equation in gas dynamics, I suspect that there is a similar procedure to reduce the IBMs to more manageable equations at the macroscopic level. Food for thought.

References:

Joseph O. Hirschfelder, Charles Francis Curtiss, Robert Byron Bird, *Molecular theory of gases and liquids*, Wiley, 1954

J. H. Ferziger and H. G. Kaper, *Mathematical Theory of Transport Processes in Gases*, North-Holland Publ. Co., Amsterdam, 1972

Sydney Chapman, T. G. Cowling and C. Cercignani, *The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases*, Cambridge Mathematical Library, 1991