Проблема необратимости и функциональная механика И.В. Волович Математический институт им. В.А. Стеклова РА

Проблема необратимости заключается в том, как совместить обратимость по времени микроскопической динамики с необратимостью макроскопических уравнений. Эта фундаментальная проблема рассматривалась в известных работах Больцмана, Пуанкаре, Боголюбова, Фейнмана, Ландау и других авторов, и оставалась открытой.
Недавно был предложен следующий подход к решению проблемы необратимости: предложена новая формулировка классической и квантовой механики, которая необратима по времени. Таким образом снимается противоречие между обратимость микроскопической и необратимость макроскопической динамики, поскольку обе динамики в предлагаемом подходе необратимы.

Широко используемое понятие микроскопического состояния системы как точки в фазовом пространстве, а также понятия траектории и микроскопических уравнений движения Ньютона не имеют непосредственного физического смысла, поскольку произвольные вещественные числа не наблюдаемы.
Фундаментальным уравнением микроскопической динамики в предлагаемом неньютоновском "функциональном" подходе является не уравнение Ньютона, а уравнение типа Фоккера—Планка. Показано, что уравнение Ньютона в таком подходе возникает как приближенное уравнение, описывающее динамику средних значений координат для не слишком больших промежутков времени. Вычислены поправки к уравнениям Ньютона.
Такой подход потребовал также пересмотра обычной Копенгагенской интерпретации квантовой механики.
I.V. Volovich, “Randomness in classical mechanics and quantum mechanics”, Found. Phys., 41:3 (2011), 516–528;
http://arxiv.org/pdf/0907.2445.pdf

Time Irreversibility Problem
Non-Newtonian Classical Mechanics
Functional Probabilistic General Relativity
Black Hole Information Paradox

Time Irreversibility Problem
The time irreversibility problem is the problem of how to explain the irreversible behaviour
of macroscopic systems from
the time-symmetric microscopic laws:
Newton, Schrodinger Eqs –- reversible
Navier-Stokes, Boltzmann, diffusion,
Entropy increasing --- irreversible

Time Irreversibility Problem
Boltzmann, Maxwell, Poincar´e, Bogolyubov,
Kolmogorov, von Neumann, Landau, Prigogine,
Feynman, Kozlov,…
Poincar´e, Landau, Prigogine, Ginzburg,
Feynman: Problem is open.
We will never solve it (Poincare)
Quantum measurement? (Landau)
Lebowitz, Goldstein, Bricmont:
Problem was solved by Boltzmann

Boltzmann`s answers to:
Loschmidt: statistical viewpoint
Poincare—Zermelo: extremely long
Poincare recurrence time
Coarse graining
Not convincing…

Ergodicity
Boltzmann, Poincare, Hopf, Kolmogorov, Anosov, Arnold, Sinai,…:
Ergodicity, mixing,… for various important deterministic mechanical and geometrical dynamical systems

Bogolyubov method
1. Newton to Liouville Eq.
Bogolyubov (BBGKI) hierarchy
2. Thermodynamic limit (infinite number of particles)
3. The condition of weakening of initial correlations between particles in the distant past
4. Functional conjecture
5. Expansion in powers of density
Divergences.

Why Newton`s mechanics
can not be true?
Newton`s equations of motions use real numbers while one can observe only rationals. (s.i.)
Classical uncertainty relations
Time irreversibility problem
Singularities in general relativity

Classical Uncertainty Relations

Newton Equation
Phase space (q,p), Hamilton dynamical flow

Newton`s Classical Mechanics
Motion of a point body is described by the
trajectory in the phase space.
Solutions of the equations of Newton or Hamilton.
Idealization: Arbitrary real numbers—non observable.
Newton`s mechanics deals with
non-observable (non-physical) quantities.

Real Numbers
A real number is an infinite series,
which is unphysical:

Try to solve these problems by developing a new, non-Newtonian mechanics.
And new, non-Einsteinian general relativity

We attempt the following solution of the irreversibility problem:
a formulation of microscopic dynamics which is irreversible in time: Non-Newtonian Functional Approach.

Functional formulation of classical mechanics
Here the physical meaning is attributed not to an individual trajectory but only to a bunch of trajectories or to the distribution function on the phase space. The fundamental equation of the microscopic dynamics in the proposed "functional" approach is not the Newton equation but the Liouville or Fokker-Planck-Kolmogorov (Langevin, Smoluchowski) equation for the distribution function of the single particle.
.

States and Observables in
Functional Classical Mechanics

States and Observables in
Functional Classical Mechanics
Not a generalized function

Fundamental Equation in
Functional Classical Mechanics
Looks like the Liouville equation which is used in statistical physics
to describe a gas of particles but here we use it to describe a single particle.(moon,…)
Instead of Newton equation. No trajectories!

Cauchy Problem for Free Particle
Poincare, Langevin, Smolukhowsky ,
Krylov, Bogoliubov, Blokhintsev, Born,…

Free Motion

Delocalization

Newton`s Equation for Average

Comparison with Quantum Mechanics

Liouville and Newton. Characteristics

Corrections to Newton`s Equations
Non-Newtonian Mechanics

Corrections to Newton`s Equations

Corrections to Newton`s Equations

Corrections

The Newton equation in this approach appears as an approximate equation describing the dynamics of the expected value of the position and momenta for not too large time intervals.
Corrections to the Newton equation are computed.
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Fokker-Planck-Kolmogorov versus Newton

Boltzmann and Bogolyubov Equations
A method for obtaining kinetic equations from the Newton equations of mechanics was proposed by Bogoliubov. This method has the following basic stages:
Liouville equation for the distribution function of particles in a finite volume, derive a chain of equations for the distribution functions,
pass to the infinite-volume, infinite number of particles limit,
postulate that the initial correlations between the particles were weaker in the remote past,
introduce the hypothesis that all many-particle distribution functions depend on time only via the one-particle distribution function, and use the formal expansion in power series in the density.
Non-Newtonian Functional Mechanics:
Finite volume. Two particles.

Liouville equation for
two particles

Two particles in finite volume

If
satisfies the Liouville equation then
obeys to the following equation
Bogolyubov type equation for two particles in finite volume

Kinetic theory for two particles
Hydrodynamics for two particles?

No classical determinism
Classical randomness
World is probabilistic
(classical and quantum)
Compare: Bohr, Heisenberg,
von Neumann, Einstein,…

Single particle (moon,…)

Newton`s approach: Empty space (vacuum) and point particles.
Reductionism: For physics, biology economy, politics (freedom, liberty,…)
This approach: No empty space. Probability distribution. Collective phenomena. Subjective.

Fixed classical spacetime?
A fixed classical background spacetime
does not exists (Kaluza—Klein, Strings, Branes). No black hole metric.
There is a set of classical universes and
a probability distribution
which satisfies the Liouville equation
(not Wheeler—De Witt).
Stochastic inflation?

Functional General Relativity
Fixed background .
Geodesics in functional mechanics
Probability distributions of spacetimes
No fixed classical background spacetime.
No Penrose—Hawking singularity theorems
Stochastic geometry? Stochastic BH?

Quantum gravity.
Superstrings
The sum over manifolds is not defined.
Algorithmically unsolved problem.

Example

Fixed classical spacetime?
A fixed classical background spacetime
does not exists (Kaluza—Klein, Strings, Branes).
There is a set of classical universes and
a probability distribution
which satisfies the Liouville equation
(not Wheeler—De Witt).
Stochastic inflation?

Quantum gravity Bogoliubov
Correlation Functions
Use Wheeler – de Witt formulation for QG.
Density operator of the universe on
Correlation functions

Factorization
Th.M. NieuwenhuizenTh.M. Nieuwenhuizen, I.V. (2005)

QG Bogoliubov-Boltzmann Eqs

Conclusions
BH and BB information loss (irreversibility) problem
Functional formulation (non-Newtonian) of classical
mechanics: distribution function instead of
individual trajectories. Fundamental equation:
Liouville or FPK for a single particle.
Newton equation—approximate for average values.
Corrections to Newton`s trajectories.
Stochastic general relativity. BH information problem.
QG Bogoliubov-Boltzmann equations.

Спасибо за внимание!