Probability, Typicality and Emergence in Statistical Mechanics

Angelo VULPIANI

Dept. of Physics, University Sapienza, Rome, Italy

Seminario delle Meccaniche, Roma, 11 Maggio 2020 (online talk)

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 1 / 41 Acknowledgments

THANKS to: Marco Baldovin, Lorenzo Caprini, Patrizia Castiglione, Fabio Cecconi, Massimo Cencini, Luca Cerino, Sergio Chibbaro, Massimo Falcioni, Annick Lesne, Andrea Puglisi and Lamberto Rondoni.

* M. Baldovin, L. Caprini and A. Vulpiani Irreversibility and typicality: a simple analytical result for the Ehrenfest model Physica A, 524, 422 (2019). * P. Castiglione, M. Falcioni, A. Lesne and A. Vulpiani, Chaos and coarse graining in statistical mechanics, (Cambridge University Press, 2008). * L. Cerino, F. Cecconi, M. Cencini, and A. Vulpiani, The role of the number of degrees of freedom and chaos in macroscopic irreversibility Physica A 442, 486 (2016). * S. Chibbaro, L. Rondoni, and A. Vulpiani, On the Foundations of Statistical Mechanics: Ergodicity, Many Degrees of Freedom and Inference Communications in Theoretical Physics 62, 469 (2014).

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 2 / 41 Why a talk on foundations of statistical mechanics?

a) Interest for itself, both for scientists and philosophers.

b) Today the development of statistical mechanics has reached the point of delving into its origin, this was already stressed by S.K. Ma in his book. For instance a challenging frontier in modern statistical physics is concerned with systems with a small number of degrees of freedom, far from the thermodynamic limit, and non Hamiltonian systems (granular material, active matter etc).

c) Pedagogical motivations: discuss in an explicit way some diﬃculties in the treatment of basic topics.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 3 / 41 The conceptual problem

The relevance of the probability theory for the statistical mechanics is obvious and cannot be underestimated. At physical, as well conceptual, level one has to face a problem: what is the link between the probabilistic computations (i.e. the averages over an ensemble) and the actual results obtained in laboratory experiments which, a fortiori, are conducted on a single realization (or sample) of the system under investigation?

In my opinion the most important role of the probability theory is to explain typicality, i.e. the (experimental) fact that a macroscopic observable in a single system is close to its average, this is an emergent property.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 4 / 41 A proposal for he link between probability and physics: the Cournot’s principle

Antoine Cournot (1801-1877) proposed the principle of impossibility: An event with very small probability will not happen. Such a principle dates back to Jakob Bernoulli (Ars Conjectandi 1713) Something is morally certain if its probability is so closed to certainty that shortfall is imperceptible

Eminent mathematicians (e.g. P.Levy, J.Hadamard, A.N.Kolmogorov) considered the Cournot’s principle as the only sensible connection between probability and empirical world.

The non abstract character of the probability: Probability is a physical property just like lenght and weight (Paul Levy)

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 5 / 41 The status of the understanding of typicality

There are (few) rigorous results in the limit N 1: * In the equilibrium problem (ergodicity problem) for weakly interacting systems and a proper class of observables (Khinchin, Mazur and van der Linden). * For the non equilibrium problem (H theorem), for diluted gases in special regime, Grad- Boltzmann limit (Lanford).

Several numerical computations and philosophical debates.

In the following I discuss a) the Ehrenfest model: in such a stochastic system it is possible to show in esplicit rigorous way that, in the limit N → ∞, the irreversibility is present for almost all the realizations, b) some simple numerical experiments of systems with many particles, showing irreversibility in a single macroscopic body.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 6 / 41 The Great Vision

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 7 / 41 Probability and real world

The two great ideas of Boltzmann:

1 Introduction of probability to compute physical quantities

2 A link between the microscopic dynamics (Hamilton equations) and the macroscopic world (thermodynamics)

The issue 2 is summarized by the celebrated relation

S = k ln W . Where S is the entropy, W is the ”volume of the phase space” of the macroscopic conﬁguration. 1 Z W (E, V , N) = δ(H(Q, P) − E)d3N Qd3N P . N!h3N

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 8 / 41 The ergodic hypothesis

Consider a macroscopic system with N interacting particles, the microscopic state is described by a X ∈ R6N whose components are the positions and the momenta of the particles. When an instrument measures some quantity, e.g. the pressure, it performs a time average, on a measurement time T , of some function of X. The instrument computes

1 Z T A¯T = A(X(t))dt . T 0

Of course for the computation of A¯T one needs the knowledge of the initial condition X(0) and the ability to determine the time evolution X(t).

This is surely beyond the human possibilities.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 9 / 41 The ingenious Boltzmann’s idea is to replace the time average with a suitable average on the the phase space. He assumed that

1 Z T Z lim A(X(t))dt = A(X)ρmicro(X)dX (1) T →∞ T 0 where ρmicro(X) is the microcanonical probability density.

This is the ERGODIC HYPOTHESIS Once the ergodic hypothesis is assumed to be valid in the limit of N 1 it is easy to derive, for a system which exchanges energy with an external environment, the Boltzmann-Gibbs distribution for the canonical ensemble.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 10 / 41 Beyond mathematics: physical ergodicity

Now, after many clear numerical evidences, starting from FPUT (Fermi, Pasta, Ulam and Tsingou) work (1955) on chain of non linear oscillators, and the KAM (Kolmogorov, Arnold and Moser) theorem on non integrable systems (1954-1963) we know that for sure the ergodic hypothesis, strictly speaking, cannot be valid.

* Why, in spite of this, is the Boltzmann-Gibbs distribution valid? * Why does the molecular dynamics give the proper results?

The Khinchin’s approach:

The ergodic hypothesis is basically true if

* N 1 * we consider ”suitable” observables * we allow for a failure of (1) in a ”small region” (i.e. with small probability)

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 11 / 41 Khinchin has been able to show that for the class of sum functions with the shape

N X f (X) = fn(qn, pn) n=1 one has

|δf (X)| c c Prob ≥ 1 ≤ 2 N N1/4 N1/4 where δf (X) is the diﬀerence between the time average starting from X and the ensemble average.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 12 / 41 Technical notes

The original work of Khinchin was only for non interacting system, i.e. X H = h1(qn, pn) n Mazur and van der Linden generalised the result to the more physical interesting case of (weakly) interacting particles. X X H = h1(qn, pn) + V (|qn − qn0 |) n n,n0

Many, but not all, macroscopic observables are sum functions.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 13 / 41 The essence of the Khinchin, and Mazur and van der Linden result

Basically Khinchin, Mazur and van der Linden showed that, although the ergodic hypothesis mathematically does not hold, it is ”physically” valid if we are ”tolerant” i.e. we accept that in system with N 1, ergodicity can fails in regions whose probability is small O(N−1/4), i.e. vanishing in the limit N → ∞.

* The dynamics has a marginal role * The very relevant ingredient is the large number of particles * The technical aspect is the Law of large numbers (which needs N 1)

On the other hand at physical level, one can say that the Khinchin result is not enough, e.g. given generic initial conditions, and observable A, it is not easy to ﬁnd the minimum value of measurement time T such that time average gives the correct result, i.e. A¯T '< A >, see diﬃculties in the FPUT problem.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 14 / 41 The old and debated problem of the irreversibility

The origin of the diﬃculties:

* The microscopic world is ruled by laws (Hamilton eq.s) which are invariant under time reversal:

t → −t , q → q , p → −p .

* The macroscopic world is described by irreversible equations, e.g. the Fick equation for the diﬀusion

∂t C = D∆C . How is it possible to conciliate the two above facts?

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 15 / 41 An aged debate

The two celebrated criticisms to the Boltzmann’s H theorem by

• Loschmidt (about reversibility) and • Zermelo (about recurrency).

Already Boltzmann and Smoluchowski understood that the criticism by Zermelo is not a problem: A process appears irreversible when the initial state has a recurrence time which is long compared to the time of observation (Smoluchowski) Actually the recurrence time is very large in macroscopic bodies (Kac lemma)

N < TR >= τoC

where τo is a typical time, C > 1 depends on the wished precision for the recurrency.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 16 / 41 A PERSONAL IDRIOSINCRASY ABOUT STATISTICAL ENSEMBLES

In standard textbooks one can read rather obscure sentences as an ensemble is an inﬁnite collection of identical systems. Such ruse, due to Maxwell, Boltzmann and Gibbs, was spectacularly successful, but it is ultimately unconvincing. See e.g. W.Zurek Eliminating ensembles from equilibrium statistical physics .... Phys. Rep. 755,1 (2018).

• A scientist dealing with a simple individual system cannot be forced to consider a collection of systems to derive properties of her/his single system. • The idea of statistical ensemble is misleading and potentially dangerous. • The statistical ensembles do not exist, in the experiments (as well as in the numerical computations) there is always a unique system with many degrees of freedom.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 17 / 41 Use and abuse of probability theory (ensembles) and entropies

Denoting with ρ(X,t) the PdF in the phase space, as consequence of the Liouville theorem, it is easy to show that Z SG (t) = −kB ρ(X,t) ln ρ(X, t) dX = SG (0) .

Note that SG can only be deﬁned over an ensemble, otherwise ρ(X, t) is meaningless. As a consequence SG (t) is accessible only in numerical experiments with systems composed by few degrees of freedom. In addition, more crucially, it is unclear how to relate SG to irreversibility because Liouville theorem implies that SG (t) must stay constant over time!

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 18 / 41 The celebrated H theorem

Deﬁne the one particle distribution function f (q, p, t) in the the so- called µ- space. In the limit of very diluted monoatomic gas, with rather subtle assumptions, Boltzmann had been able to derive an equation for the time evolution of f (q, p, t):

∂ X pj ∂ X ∂ f (q, p, t) + f (q, p, t) + p˙j f (q, p, t) = C(f , f ) ∂t m ∂qj ∂pj j j

where C(f , f ) is a bilinear integral term which takes into account the (weak) interactions among the particles of mass m. From the above equation, it is possible to show that the quantity Z SB (t) = −kB f (q, p, t) ln f (q, p, t) dq dp .

is a non decreasing function.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 19 / 41 Troubles: The ﬁrst solution to the problem of irreversibility proposed by Boltzmann clashes with the recurrence paradox, formulated by Zermelo, and with the reversibility paradox usually attributed to Loschmidt.

A physical note For the validity of the H theorem the details of the interactions are not relevant.

A mathematical remark Modern developments show that in a hard spheres systems, in the Grad’ s limit, i.e. N → ∞, σ → 0 and Nσ2 → constant where N is the number of molecules per unit of volume and σ their diameter, if the system’s initial condition is a ”good set”, f (q, p, t) basically evolves according to Boltzmann’s equation, so that the H theorem holds.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 20 / 41 The great Lanford’s result

Given the initial condition X(0) = (q1(0), q2(0), ..., qN (0), p1(0), p2(0), ..., pN (0)) one has the f (q, p, 0) and its exact evolution fe (q, p, t) which depends by X(t) = (q1(t), q2(t), ..., qN (t), p1(t), p2(t), ..., pN (t)). Lanford, under suitable assumption (Grad-Boltzmann limit), showed that up to a time t∗ = order one mean collision time, for N 1 of X(0) is typical one has fe (q, p, t) ' fB (q, p, t)

where fB (q, p, t) is given by the Boltzmann equation. Therefore with the exception of a small region, whose measure goes to zero as N → ∞, one has an irreversible behaviour.

A criticism about typicality (from some philosophers of science): why does the the Lebesgue measure (or microcanonical distribution) play a privileged role?

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 21 / 41 The increasing of SB is not just a mathematical result

In the limit of N 1 the one particle distribution function f (q, p, t) can be seen as an empirical distribution function

N 1 X f (q, p, t) = δ[q − q (t)]δ[p − p (t)] . N n n n=1

Physically Interesting aspects:

* the PdF f (q, p, t) is a well deﬁned macroscopic observable

* f (q, p, t) can be, in principle, measured in a single system, e.g. in numerical simulations, the statistical ensembles do not play any role.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 22 / 41 From mathematics to physics

It is necessary to avoid the confusion between irreversibility and relaxation of the phase space probability distribution. If a dynamical system exhibits ”good chaotic properties”, ρ(X, 0) relaxes (in a suitable technical sense) to the invariant distribution for large times t i.e. ρ(X, 0) → ρinv (X).

But this is not irreversibility.

The question, which is physically relevant, is to show that a single macroscopic system shows an irreversible behaviour, for a ”generic” initial state.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 23 / 41 Typicality and irreversibility in stochastic models

The Ehrenfest model Consider N particles, each of which can be either in one box (A) or in another (B). The state of the Markov chain at time t is identiﬁed by the number, nt of particles in A, and the evolution is ruled by stochastic laws: the transition probabilities for the state nt = n to become nt+1 = n ± 1 are n n P = , P = 1 − . n→n−1 N n→n+1 N

The state of the Markov chain nt = n, at time t, can be seen as the ”macroscopic” state of the system, the corresponding ”microscopic” configuration is defined by the (labeled) particles which are effectively in box A.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 24 / 41 The equilibrium in this model is neq = N/2. The simplicity of the model allows us to monitor the evolution of an ensemble of initial conditions starting from state n0 by computing the 2 2 2 evolution of < nt > and σt =< nt > − < nt > . N 2 t < n >= + 1 − ∆ , t 2 N 0 N 4 t N 2 2t σ2 = + 1 − ∆2 − + 1 − ∆2 . t 4 N 0 4 N 0

Where ∆0 = n0 − N/2.

One has that < nt > → neq = N/2 exponentially fast with a characteristic time τc = −1/ ln(1 − 2/N) ' N/2 and standard deviation σt goes to its equilibrium value N/2 with a characteristic time O(N)

The results for < nt > and σt are ﬁne at the level of the (ensemble) average behavior.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 25 / 41 What about a single realisation?

6 Some realisations nt vs t for N = 10 , the coloured region corresponds to hnt i − 3σt < nt < hnt i + 3σt .

1 0.002 N=106 N=106 0.9 0.622 0.001

0.8 0.618 ) / N 〉 t

/ N 0 t n 〈 n 0.7 0.614 - t

0.6 0.61 0.62 (n 0.6 -0.001

0.5 -0.002 0 0.5 1 1.5 2 2.5 3 10-2 10-1 1 10 t / N t / N

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 26 / 41 Probability and typicality

In this system it is possible to show that, if N 1, the single trajectory is also ”typical”, i.e. Prob nt '< nt > for any t ∈ [0, T ] ' 1 where T = O(N) ,

i.e. in (almost) all the trajectories nt basically behaves as the average trajectory, at least, if N is large enough.

Consider a far- from- equilibrium initial condition, e.g. n0 ' N it is possible to prove that, if N 1, until a time O(N/2), i.e. as long as nt remains far from neq each single realization of nt stays ”close” to its average.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 27 / 41 Indeed using just simple tools of the probability theory one can show that there are N → 0, and aN → 0 as N → ∞, such that |n − < n > | Prob t t < for any t ∈ [0, T ] ≥ 1 − a . N N N

−B Namely with N ∼ N with 0 < B < 1/3 −A one has aN ∼ N with A > 0 for instance for B = 0.2 one can show that A ≥ 0.2

In the limit N 1 in the overwhelming majority of the realisations of the stochastic process nt is close to < nt |n0 >. Let us note that, in spite of the worry of some philosophers of science, in this case, there is no ambiguity about the meaning of typicality.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 28 / 41 Typicality in large deterministic systems

The model

A pipe, containing N particles of mass m, horizontally conﬁned, on the left, by a ﬁxed wall and, on the right, by a wall free to move without friction (the piston), of mass M, whose position changes due to collisions with the gas particles and under the action of a constant force F . The Hamiltonian reads

P2 X p2 X H = + i + U(|q − q |) + U (q , ..., q , X ) + FX , 2M 2m i j w 1 N i i

where U is the interacting potential among the particles of the gas, Uw denotes the interaction of the particles with the wall.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 29 / 41 Sketch of the mechanical model

xˆ

yˆ F L

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 30 / 41 In the case of perfect gas, U = Uw = 0, the system is non chaotic, and it 2 is easy to ﬁnd the statistical features at equilibrium, e.g. < X > and σX .

In presence of interactions, e.g.

Uo X 1 U(r) = 12 , Uw = Uo 12 r |xi − X | i the system is chaotic, i.e. the ﬁrst Lyapunov exponent is positive, and the statistical features at equilibrium cannot be determined in an exact way.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 31 / 41 Numerical simulations (standard molecular dynamics)

The case the initial state is sufficiently far from equilibrium * |X (0) − Xeq| σeq, and it is typical (once X (0) is fixed) e.g. * the particles of the gas are uniformly distributed, * the PdF of the velocity is the Maxwell-Boltzmann distribution (with a ”wrong” temperature, i.e. different from the one at equilibrium), the evolution of X (t) exhibits an irreversible behavior.

As in the Ehrenfest model, we observe that the single trajectory is typical: far from equilibrium, ﬂuctuations are small compared to the ensemble average value.

The presence (absence) of chaos has no role.

Irreversibility is an emergent property of the a single realization as N 1 and ”proper” initial conditions.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 32 / 41 Initial conditions very far from equilibrium and N 1

(a) 500

X 400 25

15 X σ 5 300 0 2 4 6 8 10

(b) 25 500 15 X σ 5 400 0 2 4 6 8 10 X 300

200

0 2 4 6 8 10 t (x 103)

N = 1024 and X (0) = Xeq + 10σeq in a chaotic system (a) and non chaotic (b); red lines indicates X (t) for a single realization; the black lines indicate < X (t) >. Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 33 / 41 Initial conditions close to equilibrium, or small systems

82 time

X 77

72 (a) 82 time

X 77

72 (b) 0 500 1000 1500 2000 2500 3000 t 5.0

X 3.0

1.0 (c) 0 50 100 150 200 250 300 t

Initial condition X (0) = Xeq + 3σeq; N = 1024, X (t) vs t in (a) and (b); N = 4 in (c). Note the absence of irreversibility.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 34 / 41 Spreading of an ink drop

An idealized simple model:

yi (t + 1) = yi (t) + cos[xi (t) − θ(t)] ,

xi (t + 1) = xi (t) + yi (t + 1) , i = 1, 2, ...., N N X J(t + 1) = J(t) + cos[xi (t) − θ(t)] , θ(t + 1) = θ(t) + J(t + 1) i=1

Each pair (xi , yi ) identiﬁes a ”particle” (i = 1, ..., N) on the two-dimensional torus [0, 2π]X [0, 2π], for 6= 0 the particles interact via a mean-ﬁeld-like interaction, mediated by the variables θ and J.

Some particles are black (ink), the other ones are white (water); N = NI + NW where NI are the ink particles and NW are the water particles.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 35 / 41 Irreversible spreading of an ink drop

Consider the case with NI NW and the initial condition of the system with the NW particles at equilibrium (e.g. after a long integration with NW particles only), and the NI particles uniformly distributed in a small region Q0. We monitor the number of ink particles, n(t) which at time t reside in a given set Q of area A. Asymptotically, when ink is well mixed, the NI particles will distribute 2 uniformly over the torus, and n(t) will ﬂuctuate around neq = NI A/(4π ). It is instructive to compare the behavior of n(t) for a single realization with its average, computed over an ensemble of many independent realizations of the ink drop, with the water in diﬀerent (microscopic) initial conditions chosen in the equilibrium state.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 36 / 41 Q

An example of the spreading of an ink drop t = 0, 4 × 103, 2.9 × 104, 2.33 × 105 in clock wise order from top left. The NI = 3200 ink particles are initially uniformly distributed in Q0, 7 NW = 10 , = 1.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 37 / 41 8 a) 6

0 1,2 b)

0,8

0,4

0 0 1000 2000 3000 4000 t n(t)/neq vs t and < n(t) > /neq vs t, = 1. 4 6 a) NI = 8, NW = 2500; b) NI = 2.5 × 10 , NW = 10 . Note that in the last case n(t) '< n(t) > Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 38 / 41 A Summary

The idea of the statistical ensembles can be dangerous; the unique link of the probability with real world seems the Typicality: for a macroscopic observable M one has

Prob(M(t) '< M(t) >) ' 1 when N 1

Irreversibility is an emergent property for a single realization under the following assumptions:

a) A very large number of particles, i.e. N 1.

b) An initial condition very far from equilibrium,√ • in the Ehrenfest model, |n0 − N/2| N; • in piston model, |X (0) − Xeq| σeq; • in the diﬀusion model, the ink particles initially in a small region.

Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 39 / 41 REFERENCES

* M. Baldovin, L. Caprini and A. Vulpiani, Irreversibility and typicality: a simple analytical result for the Ehrenfest model Physica A, 524, 422 (2019). * P. Castiglione, M. Falcioni, A. Lesne and A. Vulpiani, Chaos and coarse graining in statistical mechanics (Cambridge University Press, 2008). * C. Cercignani, Ludwig Boltzmann: the man who trusted atoms (Oxford Univ. Press, 1998). * L. Cerino, F. Cecconi, M. Cencini, and A. Vulpiani, The role of the number of degrees of freedom and chaos in macroscopic irreversibility Physica A 442, 486 (2016). * S. Chibbaro, L. Rondoni, and A. Vulpiani, Reductionism, Emergence and Levels of Reality (Stringer-Verlag, 2014). * S. Chibbaro, L. Rondoni, and A. Vulpiani, On the Foundations of Statistical Mechanics: Ergodicity, Many Degrees of Freedom and Inference Communications in Theoretical Physics 62, 469 (2014). * G.G. Emch and C. Liu, The logic of thermostatistical physics (Springer-Verlag, 2002). Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 40 / 41 * G. Gallavotti (Ed.) The Fermi-Pasta-Ulam problem: a status report (Springer-Verlag, 2007). * S. Goldstein, Typicality and notions of probability in physics, in Y. Ben- Menahem and M. Hemmo (Eds.), Probability in physics, Springer-Verlag, 2012, pp. 59. * S. Goldstein, Boltzmann’s approach to statistical mechanics, in J. Bricmont et al. (Eds.) Chance in physics, Springer-Verlag, 2001, pp. 39. * A.I. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover, 1949). * O.E. Lanford, The hard sphere gas in the Boltzmann- Grad limit, Physica A 106, 70 (1981). * J. L. Lebowitz, Boltzmann’s entropy and time’s arrow, Physics Today 46, 32 (1993). * P. Mazur and J. van der Linden, Asymptotic form of the structure function for real systems, J. Math. Phys. 4, 271 (1963). * N. Zangh`ı, I fondamenti concettuali dell’approccio statistico in fisica, in V. Allori, M. Dorato, F. Laudisa e N. Zangh`ı(Eds.), La natura delle cose: Introduzione ai fondamenti e alla filosofia della fisica, Carocci, 2005, pp. 202. Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 41 / 41