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In my master's thesis, I worked on mathematical multi-scale models for muscle tissue. Now after finishing it, I would like to find out if one direction could be a research topic for my PhD.

At one point, a mix between a Hamilton system and a Liouville equation hoped up. For this kind of equations I would like to develop numerical schemes, but I was unable to find related publications or literature, therefore I would like to ask here for hints about similar work or publications.

The general setting.

We consider two Hamiltonian systems with Hamiltonians $H_1$ and $H_2$, which are subject to a coupling constraint $g(q_1,q_2)$. The system $H^{(N)}_2$ is a system with $N$ identical particles.

If the constraint is in such a way that we can replace $H_2$ by its Liouville equation (which respects the constraint), we get a new system, where the $N$ particles are replaced by a density of particles.

I would call the resulting equations a 'partial Liouville equations'. Since I worked on a particular system, I have not investigated yet, under which conditions it is possible to describe densities in the phase space of $H_2$ by a partial differential equation. (Below, I have sketched my concrete setting.)

In my case, the associated Lagrangian multipliers where still recoverable from the solution of the partial Liouville equation.

Main Question: Is there a name or keyword for a setting like this? Is it already discussed somewhere else?

Optional: My current question related to the system above.

Is there a definition for symplecticity of maps $$ \mathrm{Dens}(T^* \mathcal Q) \to \mathrm{Dens}(T^* \mathcal Q), $$ where $``\mathrm{Dens}"$ denotes the space of densities over the phase space $T^* \mathcal Q$? (I want to use geometric numerical integration schemes for this setting.)

Example for a "partial Liouville equation" in muscle models.

  • Micro-scale: In each sacromere (i.e. the smallest contractible sub-unit of the muscle, drawn in the picture below) a collection of small linear springs (cross-bridges) causes the contraction. I like to think of these linear springs as a particle system.

  • Macro-scale: The complete muscle is modeled as a hyperelastic body, i.e. a large Lagrangian system.

  • Coupling: A muscle deformation also deforms the sacromeres, which causes all linear springs change their extension accordingly to the contraction. The constraint can we written as $$ g_i = q_i - q_\mathrm{fiber} + const, $$ where $q_i$ denotes the extesion of a single linear spring and $q_\mathrm{fiber}$ is the length of a sacromere in the current deformation.

To define $q_\mathrm{fiber}$, we need to consider a unit length vector field on the reference space $$ N_\mathrm{fiber}: \mathcal{B} \to T \mathcal{B}. $$ Then the stretch in fiber direction is computed to be $$ q_\mathrm{fiber} := \sqrt{ N_\mathrm{fiber}^T \mathbf{C} N_\mathrm{fiber} }, $$ where $\mathbf{C} = \mathrm{D} \varphi ^T \mathrm{D} \varphi$ is the right Cauchy-Green stress tensor. The contraction rate in fiber direction is defined as $v_\mathrm{fiber} := \frac{\mathrm{d}}{\mathrm{d}t} q_\mathrm{fiber}$.

Muscle model

Partial Liouville equation

  • Meso-scale: If we replace the sacromere model of $N$ linear springs by a partial differential equation for the flow in phase space (i.e. its Liouville equation), we will get a transport equation.

  • Coupling: If we assume that the mass of a single linear spring is zero, we can still compute the coupling terms easily. I.e. the stress in the elastic solid which is caused by the linear springs.

In the end we get a system for the deformation $\varphi$ and a denisty of linear springs $\rho_\mathrm{xb}(q)$, with $q$ denoting the extension of the linear springs, $$ \ddot \varphi = \mathbf{b} + \operatorname{Div}(\mathbf P(\mathrm{D} \varphi) + \lambda(\rho_\mathrm{xb}) \mathbf{G}), \text{[hyperelasticity]}\\ \frac{\partial \rho_\mathrm{xb}}{\partial t} + v_\mathrm{fiber} \frac{\partial \rho_\mathrm{xb}}{\partial q} = 0, \quad \text{[cross-bridge equation]} $$ where $\mathbf{G}$ denotes the derivative of the constraint $g_i$. (Which is independent of $i$.) Note, that the cross-bridge equation is just a direct consequence of the constraints $g_i$, whereas the Lagrange multiplier $\lambda$ multiplier

Note: Of course, this is not the complete muscle mode, in reality cross-bridges can attach and detach, which is the contraction mechanism. The model above is only the conservative part of the system.

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    $\begingroup$ Are you perhaps looking for a ‘master equation’? $\endgroup$
    – lcv
    Oct 23, 2018 at 17:22
  • $\begingroup$ Good hint! I was not aware that 'master equation' is the general term for Liouville, Fokker-Planck or similar equations. I'll try out if the keyword helps me to find related publications! Thanks. $\endgroup$ Oct 23, 2018 at 18:03

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