The total entropy production and its three constituent components are described both as fluctuating trajectory-dependent quantities and as averaged contributions in the context of the continuous Markovian dynamics, described by stochastic differential equations with multiplicative noise, of systems with both odd and even coordinates with respect to time reversal, such as dynamics in full phase space. Two of these constituent quantities obey integral fluctuation theorems and are thus rigorously positive in the mean by Jensen's inequality. The third, however, is not and furthermore cannot be uniquely associated with irreversibility arising from relaxation, nor with the breakage of detailed balance brought about by non-equilibrium constraints. The properties of the various contributions to total entropy production are explored through the consideration of two examples: steady state heat conduction due to a temperature gradient, and transitions between stationary states of drift-diffusion on a ring, both in the context of the full phase space dynamics of a single Brownian particle.

Transfer entropy has been used to quantify the directed flow of information between source and target variables in many complex systems. While transfer entropy was originally formulated in discrete time, in this paper we provide a framework for considering transfer entropy in continuous time systems, based on Radon-Nikodym derivatives between measures of complete path realizations. To describe the information dynamics of individual path realizations, we introduce the pathwise transfer entropy, the expectation of which is the transfer entropy accumulated over a finite time interval. We demonstrate that this formalism permits an instantaneous transfer entropy rate. These properties are analogous to the behavior of physical quantities defined along paths such as work and heat. We use this approach to produce an explicit form for the transfer entropy for pure jump processes, and highlight the simplified form in the specific case of point processes (frequently used in neuroscience to model neural spike trains). Finally, we present two synthetic spiking neuron model examples to exhibit the pertinent features of our formalism, namely, that the information flow for point processes consists of discontinuous jump contributions (at spikes in the target) interrupting a continuously varying contribution (relating to waiting times between target spikes). Numerical schemes based on our formalism promise significant benefits over existing strategies based on discrete time formalisms.

No abstract

The total entropy production of stochastic systems can be divided into three quantities. The first corresponds to the excess heat, whilst the second two comprise the house-keeping heat. We denote these two components the transient and generalised house-keeping heat and we obtain an integral fluctuation theorem for the latter, valid for all Markovian stochastic dynamics. A previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal which restricts consideration of directional variables such as velocity. . Crooks and Jarzynski [8-10] then derived work relations for a variety of dynamics which held for finite times. These were followed by similar generalised relations for the entropy production associated with transitions between stationary states [11], the total entropy production [12] and the heat dissipation required to maintain a stationary state [13]. More recently the relationship between the latter quantities has been explored [14][15][16][17] resulting in a formalism involving a division of the total entropy change into two distinct terms, the adiabatic and non-adiabatic entropy productions [18][19][20], each of which obeys appropriate fluctuation relations and which map onto the house-keeping and excess heats, respectively, of Oono and Paniconi [21]. We seek to take such a formalism and generalise its scope by the explicit inclusion of both even (e.g. spatial) and odd (e.g. momentum) variables that transform differently under time reversal. In doing so we define a new quantity which obeys an integral fluctuation theorem for all time.

Ours is a harsh and unforgiving universe, and not just in the little matters that conspire against us. Its complicated rules of evolution seem unfairly biased against those who seek to predict the future. Of course, if the rules were simple, then there might be no universe of any complexity worth considering. Perhaps richness of behaviour only emerges because each component of the universe interacts with many others, and in ways that are very sensitive to details: this is the harsh and unforgiving nature. In order to predict the future, we have to take into account all the connections between the components, since they might be crucial to the evolution, and furthermore, we need to know everything about the present in order to predict the future: both of these requirements are in most cases impossible. Estimates and guesses are not enough: unforgiving sensitivity to the detail very soon leads to loss of predictability. We see this in the workings of a weather system. The approximations that meteorological services make in order to fill gaps in understanding, or initial data, eventually make the forecasts inaccurate.So a description of the dynamics of a complex system is likely to be incomplete and we have to accept that predictions will be uncertain. If we are careful in the modelling of the system, the uncertainty will grow only slowly. If we are sloppy in our model building or initial data collection, it will grow quickly. We might expect the predictions of any incomplete model to tend towards a state of general ignorance, whereby we cannot be sure about anything: rain, snow, heatwave or hurricane. We must expect there to be a spread, or fluctuations, in the outcomes of such a model. This discussion of the growth of uncertainty in predictions has a bearing on another 1) chapter contributed to R.