State space evolution beyond mechanics

Our starting point will be the assumption that the state of our system (biological, physical, chemical, or otherwise) at a given time \( t \) is represented by a collection of \( D \) generic coordinates joined in a vector \( \mathbf{C}(t) = (C_1(t), C_2(t), C_i(t), \ldots C_D(t)) \). Note this vector depends on time implicitly.

Next, we postulate the existence of a conserved property, named energy, as a function of the state variables \( E = E(\mathbf{C}(t)) \). Differentiating the energy yields \[ \frac{\diff E}{\diff t} = \sum_j \frac{\partial E}{\partial C_j} \frac{\diff C_j}{\diff t} , \] which provides an exact expression of the rate of change of the generic coordinate \( C_i \) \[ \frac{\diff C_i}{\diff t} = \left( \frac{\partial E}{\partial C_i} \right)^{-2} \frac{\diff E}{\diff t} \frac{\partial E}{\partial C_i} - \sum_{j\ne i} \frac{\diff C_j}{\diff t} \left( \frac{\partial E}{\partial C_i} \right)^{-1} \frac{\partial E}{\partial C_j} = \sum_j L_{ij} \frac{\partial E}{\partial C_j} . \] We can write the above equation in vector-matrix form \[ \frac{\diff \mathbf{C}}{\diff t} = \mathbf{L} \frac{\diff E}{\diff \mathbf{C}} = \mathbf{K} \mathbf{C} . \] This is a general equation for the deterministic evolution of any system whose state is given by a non-stochastic vector \( \mathbf{C} \). The scope of this equation of evolution is beyond mechanics because \( \mathbf{C} \) is not limited to the positions and velocities (or momenta) of particles. Note that even if we restrict the vector to \( \mathbf{C} = (\mathbf{p}, \mathbf{q}) \) the description is still more general than Hamiltonian mechanics because the equation (3) can deal with dissipative systems.

Uncertainty and stability

The above expressions are deterministic. To introduce fluctuations we can use a variational scheme to find the rate of the deviations from the average deterministic evolution \[ \frac{\diff (\delta\mathbf{C})}{\diff t} = (\delta\mathbf{K}) \mathbf{C} + \mathbf{K} (\delta\mathbf{C}) + \frac{1}{2} (\delta\mathbf{K}) (\delta\mathbf{C}) . \] Combining this expression with (3) yields for the random vector \( \mathbf{\widetilde{C}} = \mathbf{C} + \delta\mathbf{C} \) \[ \frac{\diff \mathbf{\widetilde{C}}}{\diff t} = \mathbf{K} \mathbf{\widetilde{C}} + \mathbf{\widetilde{f}} , \] for a fluctuation component of the rate given by \( \mathbf{\widetilde{f}} = (\delta\mathbf{K}) \mathbf{C} + (1/2) (\delta\mathbf{K}) (\delta\mathbf{C}) \). Using \( \mathbf{\widetilde{K}} = \mathbf{K} + \delta\mathbf{K} \) the rate can be rewriten in the suggestive form \[ \frac{\diff \mathbf{\widetilde{C}}}{\diff t} = \mathbf{\widetilde{K}} \, \mathbf{\widetilde{C}} - \frac{1}{2} (\delta\mathbf{K}) \Big [ \mathbf{\widetilde{C}} - \mathbf{C} \Big ] . \]