The justification for stochastic time evolution equation (in stochastic thermodynamics) I came across an equation in the context of stochastic thermodynamics, specifically in the paper "ensemble and trajectory thermodynamics - a brief introduction":

The time evolution of the state is described by a Markovian master equation [where $p_s$ is the probability of state $s$]:
  $$\dot p_s(t) =\sum_{s'}W_{s, s'}(t)p_{s'}(t)$$

Or in matrix form:
$$\dot p(t) =W(t)p(t)$$
However, the paper doesn't explain in a way that is clear to me (as a non-physicist) what the justification is for modelling a physical system using an equation like this. How exactly is it justified to model an open physical system that in principle behaves deterministically as one with random transition dynamics? 


*

*Are we just making a not-theoretically-justified assumption and testing the theory purely on (macro-) empirical observation? 

*Or can we actually derive based on first principles (e.g. a precise underlying Hamiltonian) that an open system behaves as if the transition dynamics have a certain random form, if we don't know the micro-state of the system's surroundings?
 A: Consider some physical system $X$ which evolves under Hamiltonian dynamics, which are always deterministic.  This system may be isolated (not coupled to any environment), so that its Hamiltonian is constant, or it may be coupled to a work reservoir but no thermodynamic reservoirs (no heat baths, chemical baths, etc.), so that its Hamiltonian changes over time.
Let $T_{t,0}(x|x_0)$ indicate the conditional probability that the system is in state $x$ at time $t$, given initial state $x_0$ at time $0$.  Since $X$ evolves deterministically, this conditional probability distribution will have the form $T_{t,0}(x'|x) = \delta(x' - f(x))$ for some function $f$.  
Now, imagine that this system is composed of two subsystems, $X=A\times B$, and that we are only interested in the dynamics of subsystem $A$.  Given some initial distribution $p(a,b)$, the conditional probability of finding  $A$ in state $a'$ at time $t$, given that it was initially in state $a_0$, is
$$
T_{t,0}(a|a_0) = \int T_{t,0}(a,b|a_0,b_0) p_0(b_0|a_0)\, db_0 \,db.
$$
In general, $T_{t,0}(a|a_0)$ will no longer be deterministic, which simply shows that once we marginalize out the effect of subsystem $B$, subsystem $A$ will evolve stochastically. However, the conditional distribution $T_{t,0}(a|a_0)$ will in general not be Markovian. Specifically, it will not necessarily obey the Chapman-Kolmogorov equation
for $0 < t' < t$:
$$T_{t,0}(a|a_0) = \int T_{t,t'}(a|a')T_{t',0} (a'|a_0) \, da'.$$ 
To guarantee that the evolution of subsystem $A$ is Markovian, we need to assume that  the following two conditions hold: 


*

*$A$ and $B$ are statistically-independent at $t=0$, $p_0(a_0,b_0)=p_0(a_0)p_0(b_0)$.

*The joint dynamics can be written as $T_{t',t}(a',b'|a,b)=T_{t',t}(a'|a,b)p_{t'}(b')$ for all $t$ and $t'$.
The second assumption means that subsystem $B$ instantly returns to some distribution, and that this distribution doesn't depend on the state of subsystem $A$.  This property characterizes subsystems which are called heat baths (or, more generally, thermodynamic reservoirs). For a heat bath, $p_t(b)$ will be the Boltzmann distribution $\frac{1}{Z}e^{-\beta H_t^B(b)}$, where $\beta$ is the inverse temperature of the bath and $H_t^B$ is the Hamiltonian of system $B$ at time $t$. In physics language, condition 2 is derived from assumptions of weak-coupling (which means that the Hamiltonian of $B$ doesn't depend on state of $A$), infinite heat capacity (which means that the inverse temperature $\beta$ doesn't depend on state of $A$), and separation of time scales (which means $B$ relaxes instantly to $p_t(b)$). 
(Note that the second requirement often can't be exactly true for Hamiltonian dynamics, but one asks that it be approximately or "effectively" true. Note also that in physics, it is usually assumed that $H_t^B$, and thus also $p_t(b)$, is time-independent, but we will no need this assumption for our derivations.)
We now show that $T_t(a'|a)$ is Markovian given the above assumptions. For convenience, let $$G_{t',t}(a'\vert a)=\int T_{t',t}(a'\vert a,b)p_{t}(b)\,db$$
refer to the evolution of subsystem $A$ from time $t$ to time $t'$, given that $p_t(a,b)=p_t(a)p_t(b)$.
Now consider the conditional probability of subsystem A at time $t'$, given some initial state $a_0,b_0$:
\begin{align}
T_{t',0}(a'\vert a_{0},b_{0}) &=\int T_{t',t}(a'\vert a,b)T_{t,0}(a,b\vert a_{0},b_{0})\,da\,db\\
& =\int T_{t',t}(a'\vert a,b)T_{t,0}(a\vert a_{0},b_{0})p_{t}(b)\,da\,db\\
& =\int G_{t',t}(a'\vert a)T_{t,0}(a\vert a_{0},b_{0})\,da
\end{align}
where in the second line we used condition 2. 
We can then marginalize out the initial state of $B$ to give the dynamics over $A$:
\begin{align}
T_{t',0}(a'\vert a_{0})&=\int T_{t',t}(a'\vert a_{0},b_{0})\,p_{0}(b_{0})db\\&=\int\Big[\int G_{t',t}(a'\vert a)T_{t,0}(a\vert a_{0},b_{0})\,da\Big]\,p_{0}(b_{0})db\\&=\int G_{t',t}(a'\vert a)G_{t,0}(a\vert a_{0})\,da.
\end{align}
Thus, $A$ satisfies the Chapman-Kolmogorov equations and its dynamics are Markovian. One can derive a continuous-time differential operator, as in the master equation formulation mentioned above, as the limit $$L_t = \frac{d}{d\epsilon} T_{t+\epsilon, t} = \lim_{\epsilon \to 0} \frac{1}{\epsilon}(T_{t+\epsilon,t} - I),$$
where we've used that $T_{t,t}=I$.
Finally, there is also the question of how to go from the continuous-state formulation I described here to a discrete-state master equation, as often used in stochastic thermodynamics. This is usually done by coarse-graining the continuous state-space into a discrete number of "mesostates". Under the assumption that the relaxation time is very fast within each mesostate (compared to the rate of transitions between mesostates), the coarse-grained dynamics over the mesostates will be well-described by a discrete state Markovian master equation.  This can be done either before or after marginalizing out subsystem $B$ (for an example, see Ch.4 in Van Kampen's Stochastic processes in physics and chemistry, 1992).
