# On the Statistical Mechanics of Life: Schrödinger Revisited

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## Abstract

**:**

## 1. Introduction

## 2. Entropy and Order

## 3. Entropy and Statistics

## 4. Correlation and Information

_{n}(with discrete values as well for simplicity), functions on Γ. For any set of values a

_{n}define the “Boltzmann entropy” (so denoted although this formula was first written by Planck [18]):

_{n}) = k log W(a

_{n})

_{n}) the number of states where the variables have values a

_{n}, that is:

_{a}and N

_{b}be the number of possible values that the variables a and b can respectively take. Let N

_{ab}be the number of possible values that the couple (a, b) can take. If there are no constraints, clearly N

_{ab}= N

_{a}× N

_{b}. But N

_{ab}can be smaller than N

_{a}× N

_{b}if a physical constraint is in place. For instance, each end of a bar of magnetic iron can be of North or South polarity, so that N

_{a}= N

_{b}= 2, but physics implies that a and b are opposite: hence only two combinations of the couple (a, b) are allowed, namely (N, S) and (S, N), so that N

_{ab}= 2 < N

_{a}× N

_{a}= 4. This leads us a formal definition of order: we say that there is order if:

_{ab}< N

_{a}× N

_{b}

_{2}N

_{ab}− log

_{2}N

_{a}− log

_{2}N

_{b}

^{3n/2}with that accessible at thermal equilibrium:

^{micro}>> ρ

^{macro}

## 5. Metastable States as Bubbles in a Foamy Phase-Space

## 6. Channels Arise from Order

## 7. The Foamy Phase Space of Biological Systems and Progression through It

## 8. Correlation in Time and Information

_{1}) and a(t

_{2}), where t

_{1}and t

_{2}are different times. Life is then first and foremost characterized by a spectacular amount of correlation across time. Recalling that correlation is information, this can be expressed in informational terms: life is characterized by a spectacular amount of preservation of information across billions of years. One key holder of this information is of course the DNA molecule and the information it encodes. Using the precise definition of information given above (Shannon’s “relative information” as physical correlation, today often denoted also “mutual” information), we can distinguish three distinct senses in which DNA molecules carry information:

- (i)
- Each single strand of a double-stranded DNA is the template of the other: given one, we can predict the other: hence it is correlated with it; hence it has information about it. (A single strand of DNA in isolation has no information, as defined above: it is just a sequence, as unique as any other sequence.) The double strand has relative information because each strand has information about the other. This is key for reproduction.
- (ii)
- DNA encodes proteins and is therefore correlated with the proteins produced: in this sense, it has information about the proteins structuring the organism.
- (iii)
- What mostly concerns us here is the third sense in which DNA molecules carry information: the entire molecule has information because it is reproduced across time—it has correlations with the past and the future. A double-stranded DNA and its descendants carry information for billions of years, through semi-conservative replication (that is, replication for which each copy comprises half its parent molecule and half a newly assembled duplicate). The macroscopic histories that contribute to biology are histories characterized by huge amount of temporal correlation: information across very long time periods.

## 9. DNA and the Statistical Underpinning of Life

- (i)
- Metabolism is a process that makes entropy grow, and as such it is directly entropically driven;
- (ii)
- Metabolism would not happen if it were not for the biochemical structure of living matter: this structure provides channels in the complex phase space of carbon chemistry from metastable states to higher-entropy ones;
- (iii)
- Inheritance is the process that allows the mechanics to be efficient in the long term, because total entropy continues to grow. This is possible thanks to long-interval correlations in time. These are given by the preservation of information, in particular in the DNA;
- (iv)
- The structure that supports metabolism grows more complex over evolution because each complexity step opens new channels to higher entropy states.

## 10. Consequences of Variability

^{3,000,000,000}when there are only 10

^{80}particles in the universe. At the end of the day it is therefore disorder (variation), rather than order, that has allowed life to develop. It is directly the disordering power of randomness, coupled with the ratchet of metastability that is the ultimate driver of evolution. The biosphere explores wider and wider regions of the spectacularly rich phase space of carbon biochemistry, allowing increasing paths for entropy to grow. This is happening on Earth’s surface because Earth has the right temperature and pressure to allow for complex carbon chemistry, and it bathes in the low entropy of the radiation of the sun.

## 11. Evolution and its Steps

- (1)
- Replicating molecules to populations of molecules in compartments (that is, cells).
- (2)
- Unlinked replicators to chromosomes.
- (3)
- RNA as gene and enzyme to DNA and protein (the genetic code leading to protein assembly).
- (4)
- Photosynthesis, which triggered a catastrophic change in the biosphere that caused a mass extinction (the Great Oxygenation Event and associated mass extinction).
- (5)
- Prokaryotes to eukaryotes.
- (6)
- Gene regulation, which enabled cellular differentiation and the development of complex multicellular life-forms.
- (7)
- Asexual clones to sexual populations, which allowed the combinatorics discussed earlier.
- (8)
- Solitary individuals to colonies (non-reproductive castes) as we see with social insects like ants and bees.
- (9)
- Neurons, which enabled organisms to coordinate activity across their cellular agglomerations and to organise specialised behaviours such as muscle contraction: leading among other things to the Cambrian substrate revolution which again caused a mass extinction.
- (10)
- Synaptic plasticity: increase and decrease in synaptic strength, or “re-weighting”, which enabled traces of neural activity to be stored in the network as memory, and allowed organisms to use the past to predict the future.
- (11)
- Internal cognitive representations, enabling organisms to simulate reality and make predictions.
- (12)
- Development of symbolic reasoning and language in humans, enabling transmission of accumulated information across time and space.

## 12. A Recent Step: Humanity

^{11}neurons with roughly 10

^{5}synapses each. If at a given time each neuron can be firing or not, this gives more than astronomical number of 2

^{1},

^{0}11 as the number of possible brain states. These states correlate themselves with the external environment (via senses), with the past states of affairs (via memory stored in the synaptic connectivity and in the dynamical network processes) and with many other brains (via language and culture), forming a powerful tool to dealing with information and elaborating it. Let us imagine for example that on average a person has a new idea at least once an hour. Then for our species novelty emerges in the mental world at least 10

^{5}times more frequently than in the biological one ([45]).

^{9}, far above the ~ 10

^{5}individuals in other mammalian species whose individuals are of our size. From a biological perspective all this sounds like an unusually strong fluctuation that the biosphere may very well have difficulty in sustaining. Complexity is not necessarily unstable, but not necessarily stable either, as we have seen with the several mass extinctions of life that have occurred in earth’s past history. The human-specific emergent properties of our species are relatively new, and could easily be unstable.

## 13. Summary

- (a)
- A key role in biology is played by the information contained in DNA. DNA strands have information about (are correlated with) one another, and about the proteins they encode. Furthermore, thanks to the reproduction allowed by their double-strand structure, their information is carried from one time to another (molecules are correlated across time). The DNA molecule has carried information across time-spans that reach billions of years. The existence of this structure is what makes the process entropically favored. Life relies on information preserved across very long time spans.
- (b)
- A second kind of information (“relevant information”) is that component relevant for survival of the correlations between the internal state of an organism with the external environment ([37,38,39,40]). This is the basis of the information massively elaborated by brains, and indirectly the basis of the information forming culture and knowledge themselves.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The intuitive understanding of the logic of the second law. The space in the picture represents all possible states of a system. If (i) there is a variable that has value L in a small L (“Low entropy”) region and value H in a large H (“High entropy”) region, and if (ii) the evolution begins in L, then it is likely to end up in H. The converse is not true: a generic evolution that begins in H likely remains in H. Hence the system evolves irreversibly from L to H but not vice versa. Note that in this example the envelope surrounding L is completely porous—there is no barrier to movement of the state from L to H or vice versa; the directionality of movement purely arises from statistics.

**Figure 2.**Intuitive representation of metastable states: the metastable state is the region L of phase space in a system for which the dynamics cannot cross the boundary of L except through a very narrow gap. A microstate will remain long trapped in the region L (the metastable state) before occasionally finding its way out towards the stable state H. The impetus to cross the channel is something that transiently lowers entropy.

**Figure 3.**(

**a**): Intuitive (oversimplified) representation of the complex phase space of living physical system: extremely numerous metastable state regions (“bubbles”) are linked to one another via channels. (

**b**): a system can wander from one bubble to a connected one and will tend to more easily make its way through the space (dark bubbles) to larger bubbles that are easier to find and harder to leave. This means that over time, the system will tend to increase in complexity (find larger bubbles).

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Jeffery, K.; Pollack, R.; Rovelli, C.
On the Statistical Mechanics of Life: Schrödinger Revisited. *Entropy* **2019**, *21*, 1211.
https://doi.org/10.3390/e21121211

**AMA Style**

Jeffery K, Pollack R, Rovelli C.
On the Statistical Mechanics of Life: Schrödinger Revisited. *Entropy*. 2019; 21(12):1211.
https://doi.org/10.3390/e21121211

**Chicago/Turabian Style**

Jeffery, Kate, Robert Pollack, and Carlo Rovelli.
2019. "On the Statistical Mechanics of Life: Schrödinger Revisited" *Entropy* 21, no. 12: 1211.
https://doi.org/10.3390/e21121211