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Where high-energy physics and statistical mechanics meet
Christian Maes
Instituut voor Theoretische Fysica
Higgs Centre Colloquium, Edinburgh – 7 April 2017
Abstract: In the past statistical physics and elementary particle physics have had many interesting interactions and various topics have been held in common. We have in mind studies of phase transitions and symmetry breaking, the renormalization group and universality, or the Gibbs formalism itself which is also the underlying structure of quantum field theory. …….
And today..??
Did we not come from the same mother and father? Nature and the Greeks Erwin Schrödinger
(about the relationship between atomism as a physical theory and the nature of mathematical objects.)
Atomic hypothesis Physical laws Unity of science
Three revolutions of 1900-1910
Where and when did we last meet? How far advanced are you now? Did you make great speed?
Twin paradox
Close encounters: 1950-1980 - Gibbs formalism - Phase transitions and symmetry breaking, cf. Brout-Englert-Higgs mechanism - Universality and renormalization group - Conformal field theory - Scaling and scale invariance
Remark about
Mass generation by nonequilibrium driving effects Urna Basu, Pierre de Buyl, Christian Maes and Karel Netočný, Driving-induced stability with long-range effects. EPL 115, 30007 (2016).
Stabilizing the metastable or even the unstable… Standard examples: -Via feedback, dynamical control
By ClarkH - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=3846878
Stabilizing the metastable or even the unstable… Much less trivial:
Stephenson-Kapitza (inverted) pendulum
R.Citro et al, arXiv:1501.05660 [quant-ph]
Probe interacting
with
nonequilibrium medium connected to
equilibrium reservoir(s)
NONEQ STAT MECH III
Time-scale separation: probe is slow, medium relaxes fast to stationary state
Stabilizing the metastable or even the unstable…
Consider many particles undergoing a Mexican-shape self-potential (2 dim).
Short range attraction to probe
Origin is unstable fixed point for probe
or
Shown: stability of origin as fixed point of probe increases with rotation amplitude of medium.
E.g. plot of effective spring constant m, second order effect
Multiple probes in short-range interaction with driven medium
medium
coupling
Statistical force on αth-probe
NEW ATTRACTOR: stable EQUIDISTANT CONFIGURATION OF PROBES
FLOW TOWARDS CRYSTAL FORMATION
Close encounters: 1950-1980 - Gibbs formalism - Phase transitions and symmetry breaking, cf. Brout-Englert-Higgs mechanism - Universality and renormalization group - Conformal field theory - Scaling and scale invariance
Still in the 1980’s: Plenty of 2dim statistical systems were “understood” by "string" methods. cf. the string-inspired conformal field theory paper by Belavin, Polyakov, Zamolodchikov 2011: Lars Onsager Prize
Reminder:
The miracle of equilibrium
Remark about possible shortcomings of Gibbs/thermodynamic/field formalism
EQUILIBRIUM... classical result: there is essentially a unique entropy, having many faces like
- related to heat [Clausius heat theorem] - related to fluctuations [Boltzmann-Planck-Einstein theory]
- being a time-increasing function [Boltzmann H-theorem]
- governing linear response [Kubo fluctuation-dissipation theorem]
- giving rise to statistical forces [Casimir effects, entropic forces].
There is no reason that such luck remains valid also outside equilibrium. Problem of nonequilibrium statistical mechanics: What are the unifying concepts, quantities, principles?
kinetics matters
But did we then lose sight of each other…?
String theory Inflation Holography Versus
Soft matter – biophysics Nonequilibrium fluctuations Active matter
And meanwhile
Second quantum revolution Computational physics
Neuro-networks-artificial Intelligence
Is there any room for statistical arguments/physics within string theory ? Short answers: yes, - other less fundamental theories could possibly be derived from it – (idea of effective action/theory) - providing correct microstates for quantum gravity (- correct entropy) - dealing with the many string vacua? - dealing with strongly correlated systems?
More recent proposals in the context of holography: Thermal states in CFT are dual to black holes in quantum gravity. (e.g. (early) Hawking-Page transition)
Gravity is emergent, no dark matter,… (Verlinde) Entanglement is key to emergence of space(-time)… Nonlocality = wormhole effect Holographic vitrification AdS/CFT duality = Jarzynski identity
"black hole phase diagram"
ER=EPR stating that entangled particles are connected by a wormhole (or Einstein-Rosen Bridge) - Maldacena-Susskind 2013
Other more direct connections between high-energy physics and thermal physics:
- Existence of Hagedorn temperature
- Unruh effect
- For example, in study of quark-gluon plasma Ex: Theory and phenomenology of quantum chromodynamics (QCD) in the high-energy high-density regime; stochastic fronts, branching random walks. the energy evolution of QCD scattering amplitudes is similar to the time evolution of a branching random with or without selection. Review paper: Statistical physics in QCD evolution towards high energies. Stephane Munier (2015).
“New techniques to extract parton densities using Monte Carlo techniques and neural networks are currently transforming the field” (webpage collider physics group)
Time to meet again ? Black hole statistical mechanics Supersymmetry and detailed balance Emergence of space-time and entanglement Fluctuating gravitational field Self-organization vs antropic principle Universe as nonequilibrium, origin of time and evolution
Conflicts ? RG and universality in nonequilibrium physics Notion of equilibrium, entropy, thermal relaxation,…
Turning to Astrophysics/Cosmology: stat.mech and thermo for evolution of universe
- Statistical methods – description of structure – e.g. correlations in galaxy catalogs from the theory of liquids.
- Fluctuation theory e.g. Brownian Motion, Dynamical Friction, and Stellar Dynamics S. Chandrasekhar Rev. Mod. Phys. 21, 383 – Published 1 July 1949: Chandrasekhar derived the Fokker-Planck equation for stars and showed that long-range gravitational encounters provide a drag force, dynamical friction, which is important in the evolution of star clusters and the formation of galaxies.
- cosmological many-body problem e.g. what decides the distribution of objects in space, and their velocities, after the memory of their initial state has faded? Would the expansion of the universe cancel the long-range gravitational mean field so that interactions effectively have a finite range. Cf. Review by Statistical Mechanics of the Cosmological Many-body Problem and its Relation to Galaxy Clustering, William C. Saslaw and Abel Yang
What would THe Statistical Mechanician In THe Street say about two or three more recent paradoxes or problems arising in questions of cosmology and high-energy physics. [Christian Maes, No information or horizon paradoxes for Th. Smiths. European Journal of Physics Plus 130, 196 (2015).]
Horizon problem (traditionally one of the major motivations for inflation theory) Flatness problem (traditionally yet another major motivation for inflation theory) Information paradox (within black hole physics, since Hawking radiation)
rather controversial
1. Rather naive or popular presentation of the Horizon problem, Flatness problem, Information paradox. 2. Give possible answers in the very standard line of statistical mechanics. (and if time allows…) 3. Go on to reflect on kinetic features in nonequilibrium physics.
Start with popularized overview of the problems:
1. Horizon problem, Wikipedia version The horizon problem is a problem with the standard cosmological model of the Big Bang which was identified in the late 1960s, primarily by Charles Misner. It points out that different regions of the universe have not "contacted" each other because of the great distances between them, but nevertheless they have the same temperature and other physical properties. This should not be possible, given that the transfer of information (or energy, heat, etc.) can occur, at most, at the speed of light.
Horizon problem, continued
When we look at the CMB it comes from 46 billion comoving light years away. However when the light was emitted the universe was much younger (300,000 years old). In that time light would have only reached as far as the smaller circles. The two points indicated on the diagram would not have been able to contact each other because their spheres of causality do not overlap.
1. Horizon problem, continued
why does the universe, particularly the microwave background, look the same in all directions? The only way for two regions to have the same conditions (e.g., temperature), is that they are close enough to each other for information to be exchanged between them so that they can equilibrate to a common state. The fastest speed that information can travel is the speed of light. If two regions are far enough apart that light has not had enough time to travel between the regions, the regions are isolated from each other. The regions are said to be beyond their horizons because the regions cannot be in contact with each other.
2. Flatness Problem
Why is the universe density so nearly at the critical density or put another way, why is the universe so flat? Currently, the universe is so incredibly well-balanced between the positively-curved closed universe and the negatively-curved open universe. Of all the possibilities from very positively-curved (very high density) to very negatively-curved (very low density), the current nearly flat condition is definitely a special case. The balance would need to have been even finer nearer the time of the Big Bang because any deviation from perfect balance gets magnified over time.
3. Information paradox Soft general version… The so called information paradox consists of multiple questions and problems related to the construction of a quantum theory of gravity. It turns out that our understanding today is rather bad, and in particular that shows up in various specific attempts that run into inconsistencies.
3. Information paradox, the more specific version Physical information seems to ‘disappear’ in a black hole, allowing many physical states to devolve into the same state, breaking unitarity of quantum evolution. From the no-hair theorem, one would expect the Hawking radiation to be completely independent of the material entering the black hole. Nevertheless, if the material entering the black hole were a pure quantum state, the transformation of that state into the mixed state of Hawking radiation would destroy information about the original quantum state. This violates Liouville's theorem and presents a physical paradox.
Hawking radiation: ρ(t) is thermal (black body radiation) ↓ Contradiction: Entanglement entropy = Shannon entropy of ρ(t) is DECREASING after some time t.
the ‘solutions’ from statistical mechanics:
(very fast to start)
1. Ad horizon problem: a) Equilibrium is typical – equal temperatures are typical. b) Relaxation to equilibrium presupposes a more special
initial condition.
2. Ad flatness problem: Make sure you use the good measure for estimating what it means to be strange/typical. 3. Ad information paradox: a) We know since Boltzmann how unitary evolutions
become effectively dissipative in a reduced description, but do not interchange limits.
b) Do not take density matrices/distributions absolutely serious.
Ad horizon problem: Phase space of classical mechanical system at fixed energy, volume and particle number. Typically, when randomly selecting a phase space point, it belongs to the phase space region of ‘thermal equilibrium,’ where macroscopic quantities take their equilibrium values.
One should not forget that the Maxwell distribution is not a state in which each molecule has a definite position and velocity, and which is thereby attained when the position and velocity of each molecule approach these definite values asymptotically.... It is in no way a special singular distribution which is to be contrasted to infinitely many more non-Maxwellian distributions; rather it is characterized by the fact that by far the largest number of possible velocity distributions have the characteristic properties of the Maxwell distribution, and compared to these there are only a relatively small number of possible distributions that deviate significantly from Maxwell’s. (Ludwig Boltzmann, 1896)
1) There is nothing special about equal temperatures. In fact, equal temperatures are typical for all regions which are solely constrained to conservation of energy. If we imagine the universe with the standard cosmology according to the FLRW geometry with at an initial time short after the Big Bang an arbitrary matter distribution with a given total energy, then we can and should expect uniform temperature all over. That is just the statement that equilibrium is typical.
2) As a matter of logic, thermalization makes the universe less special so that thermalization cannot explain specialness; the universe would have needed to be more special before.
In other words requiring thermalization is not only not needed; it is worse than useless.
Is there no dynamical content?
J.C. Maxwell:
About the proposed reduction of the second law of thermodynamics to a theorem in dynamics –
as if any pure dynamical statement would submit to such an indignity (Letter to Tait, 1876)
The truth of the second law is as in a statistical theorem,
of the nature of a strong probability… not an absolute certainty like dynamical laws. (1878)
Ad flatness problem: whether a number is truly small or how strange it is, depends on the measure, on the distribution of what is likely.
Almost every Robertson-Walker cosmology is spatially flat. The conventional formulation of the problem implicitly assumes a measure that is uniform in curvature, which seemed intuitively reasonable. But in fact the measure in the vicinity of flat universes turns out to be inversely proportional to |curvature|5/2 which is a dramatic difference. Rather than sufficiently flat universes being rare, they are actually generic.
The flatness problem really isn’t a problem at all; it was simply a mistake. In What Sense Is the Early Universe Fine-Tuned? Sean M. Carroll arXiv:1406.3057v1 [astro-ph.CO]
Ad information paradox:
1. Macroscopic steady state descriptions are by their nature approximations, valid in some limiting regimes.
E.g. Boltzmann equation E.g. Entropy production E.g. Description of magnet
2. How serious should we take the distribution or density matrix evolved from the Liouville-von Neumann evolution equation? E.g. Shannon entropy is constant over Liouville evolution; yet, the thermodynamic entropy increases and coincides with the Shannon entropy of thermal distributions. E.g. Entanglement entropy of ground states and thermal states can be very different (area law versus volume law); yet, they can easily be considered as perturbations of each other for local observables.
Hawking radiation: ρ(t) is thermal (black body radiation) ↓ Contradiction: Shannon entropy of ρ(t) is DECREASING after some time t.
Mind-boggling to Th. Smiths! In his street such statements refer to near equality of states when evaluating local observables, a few-body observables, etc..., but not to near equality of density matrices in the trace norm or near equality of their spectra (which is however what would be needed to draw conclusions about purity). In particular the Von Neumann entropies certainly are different!!
In other words, to Th. Smiths it would be like saying that the Shannon entropy of the Liouville evolved probability distribution equals the (real) thermodynamic entropy. That is certainly false, even though there is good reason to say that distribution is thermal indeed!
Similar to ‘historical paradoxes’ of statistical mechanics:
The d'Alembert paradox (1759), meaning the rigorous conclusion from
classical mechanics that birds cannot fly. (More precisely, d'Alembert saw that both drag and lift are zero in potential flow which is incompressible, inviscid, irrotational and stationary.) The resolution is of course found in the nature of viscosity and the boundary effects as explained by the Navier-Stokes equation of 1822 - 1845.
Poincaré theorem (1889) showing that no monotonically increasing (entropy) function
could be defined in terms of the canonical variables in a theory of N-body Hamiltionian dynamics. In this way he added to the so called irreversibility paradox, and indeed its solution shows that the Poincaré theorem is quite irrelevant.
d’Alembert paradox
« Form drag » par F l a n k e r — Travail personnel. Sous licence Domaine public via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Form_drag.svg#mediaviewer/File:Form_drag.svg
Back to the 19th century ? Kinetic theory? Mesoscopic theory of dissipation, entropy, entropy production, irreversible thermodynamics, relaxation to equilibrium ? NO
Rather: ambition will be to study not only dissipation but also construction and assembly – new types of phase transitions… far from equilibrium – Effects of nonequilibrium reservoirs – active media….
WELCOME TO THE 21ST CENTURY!
