# Entropy

### The link between thermodynamic entropy and information in living systems.

Stuart Kaufmann defines life in terms of the minimum (complexity) living thing, which he calls an ‘autonomous agent’ and it has the following attributes: it is an auto-catalytic system contained within a boundary (a membrane) and it is capable of completing at least one thermodynamic work cycle. This recognises that whatever living is, it must use energy to perform work: to build itself, to differentiate itself from the wider environment, to move, replicate and repair itself. All these features of the living require energy to be ‘degraded’ (for example from light into heat) in order to 'upgrade' the living form. The requirement for a thermodynamic work-cycle is another way of saying that a living organism is a kind of engine. Thermodynamics explains precisely what is meant by an engine. It is a system which pumps entropy from energy to provide a source of work. In effect this says that a living system transfers entropy from the organism into the environment (especially by degrading energy), so we can imagine an organism as a kind of entropy pump. But what is this mysterious thing called entropy and what does it have to do with information processing?

The two concepts bearing the same name ‘entropy’ are first thermodynamic entropy and second information entropy. The confusion arises because, in a development that has become infamous in the history of physics, Claude Shannon, having derived a term relevant to information transmission, was advised to name it after the earlier defined entropy of thermodynamics because it has the same mathematical form. Some have argued that the two have nothing at all to do with each other and that the simile is only a coincidence. This denies the fact that both are derived from an argument using the frequencies of number combinations to find the probability of different microstates (see below). There is then, a probability theory root to both, but they still do refer to very different things in the end.

The statistical principles underlying thermodynamic entropy are the same statistical principles as underly information entropy, but that is the only connection between them.

Thermodynamic entropy is defined by the Second Law of thermodynamics, in which it has the simple meaning of heat divided by absolute temperature: S := W / ºK. Probability comes into this because heat is in fact the sum total of the individual kinetic energies of atoms and molecules and this can be calculated using statistical mechanics, which in turn uses probability calculations. Crucially, the greater the heat, the more ways there are for a set of particles (be they atoms or molecules) to make up the total energy - this number of ways is termed the ‘multiplicity of the system’. For example, if there were just three particles and the total energy was 5 (tiny units), then, assuming only integers values, there would be 5 ways to make up this total using integer energies. If it were 10 units, there would be 14 ways. In reality it is not that simple: energies are not integers and the number of particles is, in practice, enormous (Avagadro’s number is about 6x10^23), so the multiplicity is expressed via a probability distribution. The probability distribution expresses the likelihood of total energies, given the number of ways you could make up each total energy from the individual particle speeds (the basis of their kinetic energies). For an ideal gas, the particular distribution is (fittingly) the Maxwell* distribution and it depends on the square root of temperature. To find a more thorough explanation of all this, we recommend visiting here.

* Named after one of the ‘fathers’ of statistical mechanics (and much besides): James Clerk Maxwell.

### Life as a persistent branch-system.

Entropy is defined by the Second Law of thermodynamics. Boltzmann derived the second law from statistical mechanics and ever since, this has been widely taken to mean that closed** systems tend towards their entropy maximum, simply through probability. However, this is not true; in fact the chance of entropy increase is matched by that of decrease as closed systems, over the long run are expected to fluctuate from the maximum entropy, taking random dips from it with a frequency inversely related to their magnitude. This is because all the microscopic processes of thermodynamics (molecular movements etc.) are strictly reversible, so since entropy is entirely and only dependent on them, there is no reason why entropy should always increase. The understandable mistake comes from thinking of common applications where entropy starts low and the time-course of the system’s dynamics are thought of over a relatively short span (for example a kettle cooling). It turns out, though, that these familiar small systems are best thought of as sub-systems of a larger system which is randomly dipping from maximum entropy equilibrium. Paul Davies reminds us of a footprint in the sand on a beach: it is clearly a local drop in entropy because it forms a distinct and improbable pattern that is out of equilibrium with the flat surface around it. We intuitively know that it will dissipate as its entropy rises. However, it is wrong to assume that this is because it is part of the whole beach for which entropy is always rising to a maximum. In fact the footprint was made by a process beyond the beach which caused a small part of the beach to branch-off temporarily into a separate system. Branch systems form by external influences and if left alone, they dissipate to merge back into their parent system over time. The forming of a lower entropy branch system this way, involves a more than counter-balancing increase in entropy globally. So, what of life? We can see this as a branch system in the universe, but one that is continually replenishing its low entropy by pumping out entropy into the rest of the universe, via degrading energy. Life is a continually renewing branch process.

** In thermodynamics, closed refers to a system surrounded by a boundary across which energy, in the form of work or heat, but not matter, may pass or be exchanged with its surroundings.

### Work and Life

Life is only possible if it can do work (in the technical sense used in physics). The textbook definition of work in this technical sense means the result of a force acting over a distance in a particular direction and you might ask what that has to do with living. The answer is that in thermodynamic equilibrium, matter consists of a lot of particles moving in random directions, hence with random momentum and this results in forces (for example gas pressure) that have no coherent pattern to them, so that on average (and we are averaging over huge numbers of particles) forces balance, so there is no net movement. This lack of net movement is a result of incoherence, of randomness and therefore of lack of pattern and therefore lack of embodied information. There is kinetic energy in all the incoherent movement, but in this random configuration, it cannot yield any work. For that we need some coherence to the momentum of the particles and this is achieved by limiting their scope, that is by constraining them (for example with a chamber and piston), which of course is equivalent to reducing the entropy of the system. Work is possible when there is a net direction to forces and this necessarily implies some organised pattern to the movements. Organised pattern, in turn, implies information. Once energy is organised into some information instantiating pattern, that pattern-information can be transferred to another component of the system. A thermodynamic work cycle (refering again to Stuart Kaufmann’s observation) is a process of extracting work from organised energy by transferring the information of organisation into another form, or part of the system. So, for example, constructing a protein molecule is an act of organising matter and to do it, life must sequester that organisation from energy. It is in this sense that catabolism (making body parts like protein molecules) uses energy. The energy is not used up (the First Law of Thermodynamics ensures that energy is never created nor destroyed); rather energy is disorganised, taking the information out of it to use in the catabolic process. The energy of photosynthetically active radiation (e.g. red and blue light) is more organised that that of heat radiation, so in transforming sunlight into warmth, plant leaves are supplying the necessary ‘organisation’ for constructing plant body parts. This ‘organisation’ is the information embodied in organic molecules: first in sugar (as a sort of currency), which is then transferred into a wide variety of molecules and structures built from them. Whenever energy is converted from a ‘high grade’ such as light to a ‘low grade’ such as heat, work can be extracted and used to construct something or perform a coherent action. Since living consists of a set of coherent and highly organised chemical reactions, many of which are also catabolic, there is plenty of need for work. Life would not exist without it.