7 research outputs found
Open Markov processes: A compositional perspective on non-equilibrium steady states in biology
In recent work, Baez, Fong and the author introduced a framework for
describing Markov processes equipped with a detailed balanced equilibrium as
open systems of a certain type. These `open Markov processes' serve as the
building blocks for more complicated processes. In this paper, we describe the
potential application of this framework in the modeling of biological systems
as open systems maintained away from equilibrium. We show that non-equilibrium
steady states emerge in open systems of this type, even when the rates of the
underlying process are such that a detailed balanced equilibrium is permitted.
It is shown that these non-equilibrium steady states minimize a quadratic form
which we call `dissipation.' In some circumstances, the dissipation is
approximately equal to the rate of change of relative entropy plus a correction
term. On the other hand, Prigogine's principle of minimum entropy production
generally fails for non-equilibrium steady states. We use a simple model of
membrane transport to illustrate these concepts
The free energy requirements of biological organisms; implications for evolution
Recent advances in nonequilibrium statistical physics have provided
unprecedented insight into the thermodynamics of dynamic processes. The author
recently used these advances to extend Landauer's semi-formal reasoning
concerning the thermodynamics of bit erasure, to derive the minimal free energy
required to implement an arbitrary computation. Here, I extend this analysis,
deriving the minimal free energy required by an organism to run a given
(stochastic) map from its sensor inputs to its actuator outputs. I use
this result to calculate the input-output map of an organism that
optimally trades off the free energy needed to run with the phenotypic
fitness that results from implementing . I end with a general discussion
of the limits imposed on the rate of the terrestrial biosphere's information
processing by the flux of sunlight on the Earth.Comment: 19 pages, 0 figures, presented at 2015 NIMBIoS workshop on
"Information and entropy in biological systems
The Free Energy Requirements of Biological Organisms; Implications for Evolution
abstract: Recent advances in nonequilibrium statistical physics have provided unprecedented insight into the thermodynamics of dynamic processes. The author recently used these advances to extend Landauer’s semi-formal reasoning concerning the thermodynamics of bit erasure, to derive the minimal free energy required to implement an arbitrary computation. Here, I extend this analysis, deriving the minimal free energy required by an organism to run a given (stochastic) map π from its sensor inputs to its actuator outputs. I use this result to calculate the input-output map π of an organism that optimally trades off the free energy needed to run π with the phenotypic fitness that results from implementing π. I end with a general discussion of the limits imposed on the rate of the terrestrial biosphere’s information processing by the flux of sunlight on the Earth
Open Markov Processes and Reaction Networks
We define the concept of an `open' Markov process, a continuous-time Markov
chain equipped with specified boundary states through which probability can
flow in and out of the system. External couplings which fix the probabilities
of boundary states induce non-equilibrium steady states characterized by
non-zero probability currents flowing through the system. We show that these
non-equilibrium steady states minimize a quadratic form which we call
`dissipation.' This is closely related to Prigogine's principle of minimum
entropy production. We bound the rate of change of the entropy of a driven
non-equilibrium steady state relative to the underlying equilibrium state in
terms of the flow of probability through the boundary of the process.
We then consider open Markov processes as morphisms in a symmetric monoidal
category by splitting up their boundary states into certain sets of `inputs'
and `outputs.' Composition corresponds to gluing the outputs of one such open
Markov process onto the inputs of another so that the probability flowing out
of the first process is equal to the probability flowing into the second. We
construct a `black-box' functor characterizing the behavior of an open Markov
process in terms of the space of possible steady state probabilities and
probability currents along the boundary. The fact that this is a functor means
that the behavior of a composite open Markov process can be computed by
composing the behaviors of the open Markov processes from which it is composed.
We prove a similar black-boxing theorem for reaction networks whose dynamics
are given by the non-linear rate equation. Along the way we describe a more
general category of open dynamical systems where composition corresponds to
gluing together open dynamical systems.Comment: 140 pages, University of California Riverside PhD Dissertatio