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In physics, chemistry and related fields, a kinetic scheme is a network of states and connections between them representing the scheme of a dynamical process. Usually a kinetic scheme represents a Markovian process, while for non-Markovian processes generalized kinetic schemes are used. Figure 1 shows an illustration of a kinetic scheme.
A kinetic scheme is a network of states. Each state is special, usually, has a special number, representing a specific state in the system (although repetitions of states may occur and this depends on the system). Each pair of connected states has at least one rate; a rate A_{ji} is directional and connects states i with j. Indeed, when detailed balance exists in a system, the following relation holds, A_{ji}P_{i}(t\rightarrow\infty)=A_{ij}P_{j}(t\rightarrow\infty), for every connected states i and j. (The result represents the fact that any closed loop in a Markovian network in equilibrium does not have a net flow.)
The kinetic scheme is described with a master equation: a first-order differential equation for the probability of a system to occupy each one its states at time t; written in a matrix form, we see: \frac{d\vec{P}}{dt}=\mathbf{A}\vec{P}, where \vec{P} is a column vector (where element i represents state i), and \mathbf{A} is the matrix of connections. In a Markovian kinetic scheme the connections are simply numbers (and any jumping time probability density function for state i is an exponential, with a rate equal the value of all the exiting connections). Matrix \mathbf{A} can also represent birth and death, meaning that probability is injected (birth) or taken from (death) the system, where then, the process is not in equilibrium. (These terms are different than a birth-death process, where there we have simply a linear kinetic scheme).
An example for such a process is a reduced dimensions form.
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