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Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of coupled ordinary differential equations of the form

$$ \frac{dx_i}{dt} = \displaystyle \sum_{j,k,l \in I} K^{kl}_{ij} (x_k x_l - x_i x_j) $$

where each variable $x_i$ represents the total amount of particles with velocity $v_i$ for all $i \in I$, and $K^{kl}_{ij}$ are nonnegative constants that are nonzero only if the conservation relations

$$ v_i+v_j = v_k+v_l, \ \ \ \ \ \ |v_i|^2+|v_j|^2 = |v_k|^2+|v_l|^2 $$

are satisfied.

(In other words, make the standard assumption that collisions between particles obey conservation of momentum and conservation of energy.)

My question is: was it proved already that

(i) such a system of ODEs has a specific set of positive equilibrium points (i.e., has the set of equilibrium points been fully characterized), and that

(ii) all positive solutionstrajectories converge to these equilibria?

By the way, I am not interested in "weird" cases, where there are too few collisions (or no collisions) because the set $\{v_i |\ i \in I\}$ is too small etc. Instead, I am most interested in "reasonable" cases, e.g., where the set $\{v_i |\ i \in I\}$ consists of all velocities of the form $\mathbb Z^n \cap B(0,V_{max})$, i.e., all velocities in a nice regular grid that are contained in a ball of radius $V_{max}$, and under the assumption that $V_{max}$ is large enough to allow for "a lot" of collisions to occur; for example, take $n=2$ or $n=3$, and take $V_{max} = 10$, or $V_{max} = 100$.

My guess was that this question has been answered long time ago (for all "reasonable" cases), but I looked pretty hard for references and I could not find it. By the way, I am aware that it is very easy to prove an H-theorem in this setting; but this does not (in my opinion) immediately answer my question, because (i) some nontrivial Diophantine equations need to be solved to find the set of equilibria, and (ii) the corresponding Lyapunov function is not infinite on the boundary.

Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of coupled ordinary differential equations of the form

$$ \frac{dx_i}{dt} = \displaystyle \sum_{j,k,l \in I} K^{kl}_{ij} (x_k x_l - x_i x_j) $$

where each variable $x_i$ represents the total amount of particles with velocity $v_i$ for all $i \in I$, and $K^{kl}_{ij}$ are nonnegative constants that are nonzero only if the conservation relations

$$ v_i+v_j = v_k+v_l, \ \ \ \ \ \ |v_i|^2+|v_j|^2 = |v_k|^2+|v_l|^2 $$

are satisfied.

(In other words, make the standard assumption that collisions between particles obey conservation of momentum and conservation of energy.)

My question is: was it proved already that

(i) such a system of ODEs has a specific set of positive equilibrium points (i.e., has the set of equilibrium points been fully characterized), and that

(ii) all positive solutions converge to these equilibria?

By the way, I am not interested in "weird" cases, where there are too few collisions (or no collisions) because the set $\{v_i |\ i \in I\}$ is too small etc. Instead, I am most interested in "reasonable" cases, e.g., where the set $\{v_i |\ i \in I\}$ consists of all velocities of the form $\mathbb Z^n \cap B(0,V_{max})$, i.e., all velocities in a nice regular grid that are contained in a ball of radius $V_{max}$, and under the assumption that $V_{max}$ is large enough to allow for "a lot" of collisions to occur; for example, take $n=2$ or $n=3$, and take $V_{max} = 10$, or $V_{max} = 100$.

My guess was that this question has been answered long time ago (for all "reasonable" cases), but I looked pretty hard for references and I could not find it. By the way, I am aware that it is very easy to prove an H-theorem in this setting; but this does not (in my opinion) immediately answer my question, because (i) some nontrivial Diophantine equations need to be solved to find the set of equilibria, and (ii) the corresponding Lyapunov function is not infinite on the boundary.

Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of coupled ordinary differential equations of the form

$$ \frac{dx_i}{dt} = \displaystyle \sum_{j,k,l \in I} K^{kl}_{ij} (x_k x_l - x_i x_j) $$

where each variable $x_i$ represents the total amount of particles with velocity $v_i$ for all $i \in I$, and $K^{kl}_{ij}$ are nonnegative constants that are nonzero only if the conservation relations

$$ v_i+v_j = v_k+v_l, \ \ \ \ \ \ |v_i|^2+|v_j|^2 = |v_k|^2+|v_l|^2 $$

are satisfied.

(In other words, make the standard assumption that collisions between particles obey conservation of momentum and conservation of energy.)

My question is: was it proved already that

(i) such a system of ODEs has a specific set of positive equilibrium points (i.e., has the set of equilibrium points been fully characterized), and that

(ii) all positive trajectories converge to these equilibria?

By the way, I am not interested in "weird" cases, where there are too few collisions (or no collisions) because the set $\{v_i |\ i \in I\}$ is too small etc. Instead, I am most interested in "reasonable" cases, e.g., where the set $\{v_i |\ i \in I\}$ consists of all velocities of the form $\mathbb Z^n \cap B(0,V_{max})$, i.e., all velocities in a nice regular grid that are contained in a ball of radius $V_{max}$, and under the assumption that $V_{max}$ is large enough to allow for "a lot" of collisions to occur; for example, take $n=2$ or $n=3$, and take $V_{max} = 10$, or $V_{max} = 100$.

My guess was that this question has been answered long time ago (for all "reasonable" cases), but I looked pretty hard for references and I could not find it. By the way, I am aware that it is very easy to prove an H-theorem in this setting; but this does not (in my opinion) immediately answer my question, because (i) some nontrivial Diophantine equations need to be solved to find the set of equilibria, and (ii) the corresponding Lyapunov function is not infinite on the boundary.

Source Link
DC47
DC47
  • 111
  • 5

What is known about discrete versions of the spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities?

Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of coupled ordinary differential equations of the form

$$ \frac{dx_i}{dt} = \displaystyle \sum_{j,k,l \in I} K^{kl}_{ij} (x_k x_l - x_i x_j) $$

where each variable $x_i$ represents the total amount of particles with velocity $v_i$ for all $i \in I$, and $K^{kl}_{ij}$ are nonnegative constants that are nonzero only if the conservation relations

$$ v_i+v_j = v_k+v_l, \ \ \ \ \ \ |v_i|^2+|v_j|^2 = |v_k|^2+|v_l|^2 $$

are satisfied.

(In other words, make the standard assumption that collisions between particles obey conservation of momentum and conservation of energy.)

My question is: was it proved already that

(i) such a system of ODEs has a specific set of positive equilibrium points (i.e., has the set of equilibrium points been fully characterized), and that

(ii) all positive solutions converge to these equilibria?

By the way, I am not interested in "weird" cases, where there are too few collisions (or no collisions) because the set $\{v_i |\ i \in I\}$ is too small etc. Instead, I am most interested in "reasonable" cases, e.g., where the set $\{v_i |\ i \in I\}$ consists of all velocities of the form $\mathbb Z^n \cap B(0,V_{max})$, i.e., all velocities in a nice regular grid that are contained in a ball of radius $V_{max}$, and under the assumption that $V_{max}$ is large enough to allow for "a lot" of collisions to occur; for example, take $n=2$ or $n=3$, and take $V_{max} = 10$, or $V_{max} = 100$.

My guess was that this question has been answered long time ago (for all "reasonable" cases), but I looked pretty hard for references and I could not find it. By the way, I am aware that it is very easy to prove an H-theorem in this setting; but this does not (in my opinion) immediately answer my question, because (i) some nontrivial Diophantine equations need to be solved to find the set of equilibria, and (ii) the corresponding Lyapunov function is not infinite on the boundary.