Stochastic processes in nature

In order to elucidate the content, let me start by categorizing any natural process under either of the two headings: deterministic and stochastic. Deterministic processes are the ones that we are most familiar with since our school days. Any dynamics which is described by an equation of motion that yields a unique solution corresponding to a given initial condition is said to be deterministic. Paraphrasing that, one might say that if the initial state of a system is known, its entire subsequent evolution is known, and its fate is sealed as far as the observer is concerned. Prominent examples are the Newton’s laws or equivalently the Hamiltonian dynamics, relativistic mechanics, Maxwell’s equations, Schrodinger's equation, etc. One may argue that Schrodinger’s equation comes with an inherent randomness, but the premise of that randomness is very different from what one typically refers to as “stochastic dynamics”. In fact, as long as we are content with the knowledge about the wavefunction of a particle (assumed to be a closed system) rather than its exact state, the evolution is fully deterministic.

What are the examples of deterministic processes in nature? Well, to be precise, nothing! Unless, of course, we are dealing with a highly idealistic process that occurs at absolute zero, or with the equally unlikely proposition that a system is completely isolated from its environment, all natural processes involve stochasticity. To a very good approximation, however, the dynamics of a massive body can be described by deterministic equations, because the thermal energy of the body is astronomically small compared to changes in its mechanical energy. Nevertheless, it would be unfair on my part to substantiate the erroneous idea that stochasticity can only be of thermal origin. In fact, the ubiquity of stochasticity in a vast range of systems, covering as seemingly unrelated fields as finance, biology, computational algorithms, finance and astronomy, speaks volumes about why a researcher cannot avoid a confrontation with this concept. In fact, we can easily perceive that the sequence of activities of a human being in a day, under the same external conditions, is subject to a great deal of randomness.

Now, let’s turn our attention to the central idea of this article: a stochastic process. The obvious examples are Brownian dynamics and chemical kinetics. Neither the exact position of a Brownian particle (particles having dimensions of the order of micrometres or less, that appear to undergo random motion in a medium), nor the exact time instant of the next reaction, can be predicted with absolute certainty, in stark contrast to Newton’s laws which can predict the position of a projectile in space (at a given time) with uncanny precision. Nevertheless, predictability is not entirely washed out, even though its limits are revised. Thus, we can still compute the probability of the position of the Brownian particle at a given time, or the rate of reaction between two reactants. Similarly, the probability of a given force acting on a star in a large stellar system at a given time can be predicted, although it would be a humongous (read “impossible”) task to predict the exact value of this force.

Now we would try to devote some time to take a closer view of Brownian motion. It is a simple extrapolation of what is known as the random walk problem in physics. The problem is as follows: an inebriated man is left with the task of finding his way home, all by himself. Since the light of reason has deserted him, he is equally likely to take a step towards left or towards right. The obvious question would be, how long he would take to reach his destination. Well, it can be shown, that he will do so eventually, given enough time, although the average time taken will diverge with the system size (if he is walking on a lattice, then the dimensions of this lattice quantify the system size). Results become non-trivial in three dimensions, and there will be a finite probability of the man never being able to make it. Next, if one keeps reducing the step size of the man till it reaches scales much smaller than the resolving power of our microscope, one readily obtains the statistics of a Brownian particle.

A property of such a random walk, as has been outlined above, is that of the first passage time. It is the time interval after which a random walker first reaches a given location. A suggestive diagram on the cover page of Sydney Redner’s monograph titled “A Guide to First-Passage Processes” metaphorically explains the concept. There, a preoccupied man is found to be walking towards the edge of a cliff, with a crocodile staring upwards from the waters below. Obviously, he is destined to be obliterated once he reaches the edge and topples over for the first time.

The concept is often exemplified by the famous Gambler’s ruin problem. Let us consider two gamblers, A and B, who have finite amounts of money at their disposal, and partake in a gamble as long as one of them is not ruined (i.e., loses all his money). A fair coin is flipped, and each time the outcome is heads, gambler A receives a penny from his opponent, and the reverse happens if the outcome is tails. It can be shown that in the long run, the one who has higher number of pennies to begin with, ends up on the winning side - a principle that is routinely used in the casinos.

The applications of random walk are numerous. Google’s page ranking algorithm (an algorithm that helps in enlisting pages in the order of decreasing relevance), for instance, is heavily dependent on random walk statistics. In social networks, similar ideas are invoked in order to suggest possible friends to the person in question. Another useful application is the inference of the input to a black box, when we are aware of the outputs, known as a search algorithm. The Grover’s algorithm is supposed to carry out an analogous job for a quantum computer, which has immensely boosted research directed towards the study of quantum random walks.

Now, let us “open” the boundaries of the system, which means that random walkers can emerge on the lane from a source (a reservoir that can furnish an unlimited supply of some stuff) at one end, and can leave the lane through a sink (another reservoir that can absorb any amount of stuff that happens to reach its territories) at the other end. The rates of ingress and egress are not equal in general. Of course, we are now dealing with multiple random walkers, rather than one. Suppose that the walkers have partially regained their sanity, and do not take entirely random steps, but are biased to move in one direction. Further, we can impose the constraint that they are reluctant to bump into each other (technically, this means that two particles cannot be simultaneously present at the same lattice site). With all these suppositions in place, we arrive at what is known as the asymmetric simple exclusion process (ASEP) in the physics literature. Such processes have been used to qualitatively understand the behaviour of traffic on a street, or of molecular motors on microtubules inside a biological cell, and the conditions under which one might expect roadblocks.

There are tons of other branches that thrive on stochasticity. The fields of pattern formation/self-organization (like crystallization of molecules, in which an organized structure emerges in the absence of any external agent) and synchronization problems (like randomly flashing fireflies eventually reaching a steady state where they flash in synchronization with each other) are two such areas that are replete with stochastic events. Gene expression in cells also shows considerable amounts of stochasticity so as to attract the interests of physicists and mathematicians. Biological motors, that help in intracellular cargo transport and a host of other activities, rectify thermal noise using chemical energy and thus can undergo unidirectional motion. Stochastic differential equations can be used to model the evolution of the prices of stock options. The Black-Scholes model laid the foundations for such analysis for the European financial markets, whose impact was deservedly acknowledged in the form of the 1997 Nobel Prize in Economics. Concepts of polymerization reactions, that involve the aggregation of two short polymers to produce a longer one, or the fragmentation of a long polymer into shorter ones, finds its applications in the explaining the formation of stars. Game theory is another area that has lent itself to a wide spectrum of applications: in economics, business, political science, biology, psychology, computer science, etc. The arenas enlisted above form only the tip of an iceberg. It is up to the reader to fathom and explore the vastness of the field.

What, then, is the moral of the story? It apparently seems that nature is favourably skewed towards phenomena that involve some degree of randomness in their dynamics. Why things are geared up against absolute determinism is more of a philosophical question. Determinism makes a process easier to understand and allows absolute predictability, but we would all agree that it is stochasticity that makes life far more interesting and imparts enough richness to the events that take place all around to keep us motivated to unravel the mysteries of nature.