There are a lot of non-equilibrium processes examples given in physics literature. But some processes that are present in everyday life are not treated.

As an example, the formation of pudding can be considered. Suppose one has two ingredients for pudding making: one liquid an one binder. After mixing these substances and heating the mixture one gets a pudding.

Let be $f(t,\Gamma \otimes \Xi )$ a phase space distribution of this 2-component system. Here, $t$ denotes the time, $\Gamma$ is the space of coordinates and momenta of all the particles of the system and $\Xi$ is a process-specific variable that determines whether the binder begins to bind liquid around the binder to the pudding form. More precisely if $\sigma \in \{ 0,1\}$ where $\sigma = 0$ means that no binding takes place and $\sigma = 1$ means that binding takes place then one can define a phase space

$f(t,\Gamma \otimes \Xi ) = f(t, q_{i,liq}, p_{i,liq}, q_{i,b}, p_{i,b}, \sigma_{i,b})$.

One can formulate the Lioville equation:

$\partial_tf + \sum_{i=1}^{N_{liq}} (\dot{q_{i,liq}}f\partial_{q_{i,liq}}f + \dot{p_{i,liq}}f\partial_{p_{i,liq}}f) + \sum_{\sigma_i}\sum_{i=1}^{N_{b}} (\dot{q_{i,b}}f\partial_{q_{i,b}}f + \dot{p_{i,b}}f\partial_{p_{i,b}}f)|_{\sigma_i} = C(f)$.

The exchange term $C(f)$ describes the transition from non-binding to binding. There can be assumed a poisson-distributed transition ors something similar. Clearly, also the forces $\dot{p_{i,liq}},\dot{p_{i,b}}$ depend on the state variables $\sigma_i$ to model the binding.

Question: Is my method plausible? Are there some similar problems treated in literature (i.e. phase space density functions with additional process-characteristic parameters)?


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