Thermodynamics Beyond Molecules

Thermodynamics Beyond Molecules

Thermodynamics and the Evolution of Populations

Themis Matsoukas

Department of Chemical Engineering Pennsylvania State University [email protected]

International Conference on Thermodynamics 2.0 June 22-24, 2020, Worcester, MA

INTRODUCTION

• _{Thermodynamics is auniversallanguage.} What does this mean?

• _{The impulse to apply thermodynamics to} all sort of problems is undeniable. What drives thisintuitive urge?

• _{What are the rules for applying} thermodynamics outsidephysics?

• _{Toexportthermodynamics to the}

non-physical world we mustseparateit from physics. Is that even possible? • _{Whatisthermodynamics?}

https://phys.org/news/2018-03-glass-transition-driven-thermodynamics.html

Thermodynamics Beyond Molecules

Our goal is to:

•

_Deconstruct

_{thermodynamics}

•

_Throw

_{physics away}

•

_Reconstruct

_{“thermodynamics” without}

Thermodynamic entropy

Before Boltzmann and Gibbs thermodynamics was aboutheatandworkand its main question was tomaximizethe amount of work in a cycle. This is a variational problem and it is answered by themaximization of the entropy:

maxS(E, V, N)

Entropy function:elusive but computable.

S(E, V, N) =Z dQrev T

S

Statistical Entropy

WithBoltzmannandGibbsentropy becomes a

functionalof a probability distribution:

S[p] = −X i

pilogpi

The equilibrium distribution maximizes the entropy functional among all feasible

probability distributions that can be assigned to a set of microstates:

p∗_{: S[p}∗_{] ≥S[p]; p ∈ E} X,Y···

Afeasibledistribution is one that satisfies all

macroscopic specifications of state (X, Y · · · ).

Ludwig Boltzmann

How Statistical Thermodynamics Works

In the canonical ensemble (¯E, V, N) the set E offeasibledistributions is all distributions that satisfy

X i pi=1, X i Eipi= ¯E

The variational problem is to maximize − P_ipilogpiunder the constraints of the feasible set and the solution is

p∗ i =e −βEi Q(β), (1) with ¯ E = −∂ logQ ∂β (2) Ω(E) = log Q − β¯E (3) β = ∂ log Ω(¯E) ∂¯E (4)

Shannon and Jaynes

Shannonobtained the entropy functional by

means unrelated to physics. Shannon’s entropy is a measure ofuncertainty. The canonical probability distribution has the maximum uncertainty among all distributions with the same mean.

Jaynestook this further:

[w may have now reached a state where sta-tistical mechanics is no longer dependent on physical hypotheses, but may become simply an example of statistical inference (Jaynes, 1957)

Claude Shannon

Thermodynamics Outside Physics

Shannon unlocked the door and Jaynes kicked it open. Entropy has escaped!

Physics Only

Entro

py

Hello World!

Random Sampling

Start with agenericprobability distribution

P(X = xi) =qiand collect arandom sample (n₁,n₂· · ·nN)of size N. Form the intensive empirical distribution of the sample pi=ni/N. The probability to obtain {p} is

P({p}) = N!Y i

qni

i

ni! Take the log:

logP({p}) N = − X i pilogp_qi i

TheMost Probable Distributionof the empirical

sample is the distribution that is being sampled {p}. 6 6 5 5 5 5 5 1 1 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 1 1 2 2 2 2 2 4 1 4 3 1 5

Biased Sampling

Bias the selection of the sample by a homogeneous functional W:

logW({p})=X

i

∂ logW({p}) ∂p_i

The probability to obtain distribution {p} in the sample is logP({p}) N = − X i pilog_wpi iqi − log(norm.) Most Probable Distribution:

p∗ i =wip_ri

Setting wi=fi/pi,anydistribution fibecomes the Most Probable Distribution

6 6 5 5 5 5 5 1 1 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 1 1 2 2 2 2 2 4 1 4 3 1 5

Space of Distributions

• _{Biased sampling from any probability} distribution {q} establishes aspace of

distributions{p} in the same domain and a

probability measureP({p}).

• _{By proper selection of the the bias,any} distribution in this space can be obtained as the Most Probable Distribution.

• _{When the sample size is large the Most} Probable Distribution isoverwhelmingly

more probablethan all others.

space of distributions {p} P({p})

Microcanonical Space

Draw biased samples from theexponential distribution

qi= e −xi/¯x

¯x

The probability of empirical distribution {p} is logP({p}) N = − X i pilog_wpi i − log ω . = log ρ({p}) | {z } microcanonical functional The Most Probable Distribution is

p∗ i =wie −βxi q with log ω = β¯x + log q ¯ x = d log q dβ , β = d log ω dβ , ρ({p∗}) =0.

Variational Condition

All distributions satisfy log ω ≥ −X

i

pilog_wpi i with the equal sign for pi=p∗i. This variational condition defines the rank of distributions in the ensemble and identifies the Most Probable Distribution by

log ω = −X i p∗ i log Results ρ = −X i pilogpi wi − log ω ρ({p}) ≤ 0 p∗ i =wie −β_xi q log ω = β¯x + log q ¯ x =d log q dβ β =d log ω dβ

Contact with Physics

Take xito be the energy level of an (N, V) system:

x_i → E

β → 1/kT

wi → number of microstates with energy E

=thermodynamic entropy

ω → microcanonical partition function

q → canonical partition function

ρ({p}) ≤ 0 → second law

All thermodynamics is recovered

Results ρ = −X i pilogpi wi − log ω ρ({p}) ≤ 0 p∗ i =wie −β_xi q log ω = β¯x + log q ¯ x =d log q dβ β =d log ω dβ

Geeralized Thermodynamics

• _{Anydistribution f (x) defined in (0 ≤ x < ∞) can be obtained by} biased sampling of the exponential distribution using a suitable bias.

• _{Anydistribution f (x), (0 ≤ x < ∞) satisfiels the familiar} thermodynamic relationships under a suitable bias.

• _{Generalized Thermodynamicsis the probabilistic calculus of the}

Most Probable Distribution. The “ensemble” is the set of all

feasible distributions.

• _{TheGeneralized Second Lawstates that the Most Probable}

Distribution is more probable than all other distributions in the feasible space.

• _{Generalized Thermodynamics applies to probability distributions} and by extension toStochastic Processesin general.

Stochastic Process

Astochastic processessamples distributions of

its stochastic variable viatrajectoriesin phase space.

Afeasible distributionis any distribution that

can be reached from fixed initial state.

Themicrocanonical spaceis all distributions

that are reached in a fixed number of transitions.

distribution

The bias W is fixed by thetransition probabilitiesalong trajectories.

TheMost Probable Distributionis the observableprobability

Evolution

Evolutionis the dynamic transformation of a population clustered by traits, n = (n1,n2· · · )whose

members undergo tranisitons of the form α0iXi+ α0jXj+ · · · → αkXk+ αlXl· · · The feasible space is all distibutions reached in G steps from fixed initial state. The probability of distribution is

P(n) =n!W(n)

Ω

with n = N!/n1!n2! · · ·. The most probable

distribution is n∗ i N =wi e−βi q

The key to unlocking the thermodynamics of the population lies in thetransition probabiities

Scieintific Reports 5 8855, 2015.https:dx.doi.org/

Concluding Remarks

Thermodynamicsisa universal language: It is the language of probability distributions and stochastic process and we have just

began to decipher its grammar T. Matsoukas, Generalized Statistical Mechanics, Springer 2018;

https://doi.org/10.1007/978-3-030-04149-6