Statistical Physics of Adaptation

Whether by virtue of being prepared in a slowly relaxing, high-free energy initial condition, or because they are constantly dissipating energy absorbed from a strong external drive, many systems subject to thermal fluctuations are not expected to behave in the way they would at thermal equilibrium. Rather, the probability of finding such a system in a given microscopic arrangement may deviate strongly from the Boltzmann distribution, raising the question of whether thermodynamics still has anything to tell us about which arrangements are the most likely to be observed. In this work, we build on past results governing nonequilibrium thermodynamics and define a generalized Helmholtz free energy that exactly delineates the various factors that quantitatively contribute to the relative probabilities of different outcomes in far-from-equilibrium stochastic dynamics. By applying this expression to the analysis of two examples—namely, a particle hopping in an oscillating energy landscape and a population composed of two types of exponentially growing self-replicators—we illustrate a simple relationship between outcome-likelihood and dissipative history. In closing, we discuss the possible relevance of such a thermodynamic principle for our understanding of self-organization in complex systems, paying particular attention to a possible analogy to the way evolutionary adaptations emerge in living things.

Popular Summary

Humans are generally accustomed to explaining the many impressive functional adaptations exhibited by living things in Darwinian terms. In other words, heritable traits that help a creature survive and reproduce in a given environment get passed along to future generations. However, from the perspective of physics, we may also note that living things are universally good at latching onto sources of work energy in their environments (e.g., sunlight) and using that energy to dissipate heat into the surrounding air or water. We can accordingly consider the general situation of matter being driven by a source of work energy in contact with a heat bath. In doing so, we can demonstrate aspects of the physical properties of such a system that might help us to understand biological organization in a new light.

We put forth here a theoretical argument for a general tendency in driven matter to adopt shapes that have special histories of extra absorption of work energy from the environment. In particular, we consider two examples: a particle hopping around an energy landscape and a population of objects that self-replicate in an exponential fashion. We numerically show that a particle’s (or population’s) likely direction of evolution is correlated with the corresponding amount of absorption and dissipation of work. Put another way, we show that the probability of a particle or population ending up in a given macroscopic state is proportional to the thermodynamic flux that occurs on the way to that state. We are able to suggest a new thermodynamic perspective on self-organization far from thermal equilibrium that draws connections between the biological world and other systems governed by the same general physical principles.

We expect that our findings will pave the way for experimental studies of the nonequilibrium physics of lifelike behavior.

Figure 1

Three separate microstates initially of equal energy $E=0$ are arranged on a line so that stochastic transitions may take place between adjacent locations separated by activation barriers of height $\mathrm{\Delta }E$. When state $x_1$ and the barrier separating it from $x_2$ are driven, such that their energies oscillate in time, a preference for $x_1$ over $x_3$ as a destination for finite time evolution from $x_2$ develops. This statistical drift is necessarily accompanied by the conversion of absorbed work into dissipated heat in the surroundings. The energies of the driven states are drawn at their extremal values in blue, with dashed lines and solid lines, respectively, in phase with each other.

Figure 2

An arbitrary nonequilibrium macrostate for the system of interest is constructed as the probability density $p_i(\mathbf{x})d\mathbf{x}$ over the microstates $\mathbf{x}$ when the system is prepared by a controlled procedure to have some macroscopic property $\mathbf{I}$. The system is externally driven for a period of duration $\tau$ by a time-varying field $H_{\text{sys}}(\mathbf{x},\lambda (t))$ while in contact with a thermal bath of inverse temperature $\beta$ that absorbs heat $\mathrm{\Delta }Q$. Afterwards, one may ask whether the system now has a new macroscopic property $\mathbf{II}$; if so, the new probability distribution over microstates is implicitly given by $p_f(\mathbf{x})d\mathbf{x}$.

Figure 3

In this scenario, we consider a starting nonequilibrium macrostate $\mathbf{I}$ that is driven by an external field while it evolves stochastically in contact with a thermal bath. We may ask whether, after some finite time $\tau$, it is more likely to be found in one of two other possible macrostates $\mathbf{II}$ and $\mathbf{III}$. The relative likelihood of these two different possibilities will be determined in part by the statistical distributions for possible values of the external entropy production $\beta \mathrm{\Delta }{Q}_{\mathbf{I}\to \mathbf{*}}$ in each case.

Figure 4

In the driven scenario depicted schematically here, two states $x_1$ and $x_2$ are separated by two different barriers representing two different possible paths for transiting between the states. Because the barriers are driven out of phase with each other, and since $\beta \mathrm{\Delta }E\gg 1$ and $r\ll 1/\tau \ll \omega$, the return probability and average dissipation for transitions from $x_2$ to $x_1$ are independent of $\mathrm{\Delta }E$, the choice of which should still affect the drift rate from $x_2$ to $x_1$ as well as the fluctuations $\mathrm{\Phi }$ for the pair of paths. As in Fig. 1, the energies of the driven states are drawn at their extremal values in blue, with dashed lines and solid lines, respectively, in phase with each other.

Figure 5

Lines of fixed reversal probability based on Eq. (18), with $g_A=1$, $g_B=2$, $\tau =10$, and $\delta =e^{-100}$. Each line is colored to show the entropy produced while generating each ending state from the initial state $N_A=N_B=1$, relative to that of the state on the line where $N_B=N_A$. On each of these surfaces of fixed reversal probability, the most likely outcome by orders of magnitude is one where $N_B\gg N_A$.