Ballistic resonance as breaking of second law: TGD view point

The popular article " Scientists have discovered a new physical paradox" (see this) tells about the work of Vitaly A. Kuzkin et al published in Phys Rev E (see this) as article with title " Ballistic resonance and thermalization in the Fermi-Pasta-Ulam-Tsingou chain at finite temperature" . The article describes very interesting experimental findings, which could provide a direct application of zero energy ontology (ZEO) based theory of self-organization.

The findings and their explanation provided by experimenters

Researchers from the Peter the Great St. Petersburg Polytechnic University (SPbPU) have discovered a new physical effect: the amplitude of mechanical vibrations can grow without external influence in which system converts its thermal energy to mechanical energy. The phenomenon is known as ballistic resonance. The description of the phenomenon involves also an abnormally high heat conductivity - one speaks of ballistic heat conductivity.

The electromagnetic analogy is very high electric conductivity: the work of Bandyopadhyay related to effects of oscillating voltage on currents flowing along microtubules demonstrates ballistic conductivity possibly reflecting underlying super-conductivity (see this).

This behavior seems to be in conflict with second law of thermodynamics telling that the vibrations should be attenuated. The researchers propose also a theoretical explanation of this paradox (see this) based on a model assuming ballistic heat conduction. One can of course wonder whether the notion of ballistic heat conduction is consistent with second law in its standard form.

Fermi-Past-Ulam-Tsingou problem (see this) relates to the finding about a theoretical model of a vibrating string with a non-linear dynamics. The expectation was that the situation develops ergodic so that energy is evenly divided between the modes of the string. It however turned out that the behavior was essentially periodic. The model explaining the behavior relies on solitons assignable to Korteveg-de-Vries equation. This phenomenon is different from the ballistic resonance observed in the experiments. In Korteveg de-Vries equation there is no dissipative term and the unexpected phenomenon is that wave pattern preserves it shape. Dissipation without energy feed would attenuate the wave.

ZEO based model for the findings

TGD suggests that a genuine explanation requires a profound change in the thinking about time- in particular the relationship between geometric time and experienced time must be updated. I call the new conceptual framework zero energy ontology ZEO) (see this). The identification of these two times in standard ontology is in conflict with simple empirical facts, and leads to a paradox related to state function reduction (SFR) taking place in quantum measurement. The non-determinism of SFR is in conflict with the determinism of Schrödinger equation.

1. According to ZEO in ordinary state function reduction (SFR) the arrow of time subsystem changes: this solves the basic paradox of quantum measurement theory. The experiments of Minev et al (see this) give impressive experimental support for the notion in atomic scales, and sow that SFR looks completely classical deterministic smooth time evolution for the observer with opposite arrow of time. This is just what TGD predicts. Macroscopic quantum jump can occur in all scales but ZEO takes care that the world looks classical! The endless debate about the scale in which quantum world becomes classical would be solely due to complete misunderstanding of the notion of time.
2. Non-standard arrow of time forces a generalization of thermodynamics. For time reversed system generalized second law applies in reverse direction of time. Dissipation with reversed arrow of time extracts energy from environment, in particular thermal energy from internal thermal environment. The energy feed necessary for self-organization reduces to dissipation in reversed arrow of time.

   This explains why self-organization is possible (see this). Standard form of the second flow would imply that also energy flows between systems go to zero: this would mean thermodynamical equilibrium everywhere - heat death. This has led to desperate theoretical proposals such as life as gigantic thermodynamical fluctuation. The recent empirical understanding suggests that this giant fluctuation would have occurred in the scale of the entire Universe and continue forever!

3. Macroscopic quantum coherence is however a necessary prerequisite for macroscopic effects. TGD predicts hierarchy of phases of ordinary matter residing at magnetic body (MB) of the system with value of effective Planck constant h_eff= nh₀ (h=6h₀) of h_effbehaving like dark matter and controlling ordinary matter (see this).. The larger the value of h_eff, the longer the scale of quantum coherence scale at MB. MB acts as master for ordinary matter in the role of slave and induces coherent behaviour. This gives rise to self-organization.

This picture could explain the observations of self induced resonance using thermal energy. A subsystem or its MB in time reversed mode would extract the thermal energy. There are many other applications. The phenomenon of stochastic resonance in which system extracts energy from external noise could have explanation along these lines. Stochastic resonance plays an important role in sensory perception by making possible amplification of weak signal in large background. There is evidence for it even in astrophysical scales. In biology metabolic energy could be extracted from metabolites and maybe also from thermal energy by time reversed dissipation by some subsystems related to metabolism.

TGD picture does not exclude the possibility of delicate models mimicking this behavior in the framework of thermodynamics. The basic challenge in this kind of effective model is to describe the presence time reversed dissipation inducing self-organization and the presence of dark matter at magnetic body phenomenologically. Energy feed as parameter gives rise to states far from thermodynamical equilibria.

For instance, the thermodynamics of ion distributions inside and outside cell is far far from thermodynamical equilibrium and and non-equilibrium thermodynamics has been developed for the modelling of this kind of systems utilizing the notions of ionic pumps and channels. The phenomenological description introduces chemical potentials as parameters to describe the non-equilibrium situation in the framework ordinary thermodynamics. Chemical potentials would model the neglected presence of h_eff>h phases of dark matter at magnetic body of the system.

See the article Ballistic resonance and zero energy ontology or the chapter Zero Energy Ontology and Matrices.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Is there a chronon of time?

This posting was inspired by a popular article telling about the proposal that Planck length is not the fundamental length as often believed but the fundamental chronon is much longer.

The idea about fundamental unit of time - usually assumed to be given by Planck time about 10^-44 seconds - is rather naive if taken to mean discretization of time. The article proposes a variant of this idea and identifies the fundamental chronon as a fundamental periodicity of dynamics and deduces for it a value about 10^-33 seconds from observed bounds to the variation of dynamical periods. This value is by 11 orders of magnitude longer than Planck time. Personally I am a little bit skeptic about reliability of these bounds since very short times are involved.

Planck time is deduced from a mere dimensional analysis argument by feeding in speed of light c, Planck constant h, and Newton's constant G so that it is rather ad hoc noton. Therefore it has been surprising to me at least how seriously people have taken it. Moreover, Planck length appears in theories - in superstring theory in particular - typically as an ad hoc formal parameter with no direct geometric interpretation. In general relativity this leads to non-renormalizability and it is not possible to quantize gravitation in this framework. In superstring theories it led to landscape catastrophe: even a smallest change in the physics at Planck length scales changes completely the physics at long length scales: butterfly effect in the theory space.

For these reasons the question whether there exists some fundamental length-/time unit or several of them is a key problem of recent day physics. Could there be some fundamental length scale or possibly several of them with a clear geometric interpretation? In TGD Universe this is indeed the case.

1. Planck length is derived quantity and CP₂ length scale defines the fundamental length, which from p-adic mass calculation for electron mass roughly 10⁴ times longer than Planck length. Space-time is continuuous but CP₂ length serves as a fundamental unit of length, kind of length stick.
2. p-Adic length scale hypothesis (PLH) predicts actually infinite hierarchy of length/time units as p-adic length scale hypothesis stating that these units are proportional to sqrt(p), p preferred p-adic prime. p-Adic length scale hypothesis in this general form emerges both from M⁸-H duality and p-adic mass calculations.
3. A stronger form of PLH states that certain primes near powers of 2 are physically favored so that in the most general case one obtains a hierarchy of length and time units coming as half octaves. This form of hypothesis not well-understood although it conforms with period doubling in chaotic systems. Also powers of other small primes are possible and there is some evidence for the powers of 3. This would relate the preferred length scales of physics in long scales to CP₂ scale.
4. TGD predicts second length scale hierarchy corresponding to the hierarchy of effective values h_eff=nh₀ of Planck constant (h=6h₀) labelling phases of ordinary matter behaving like dark matter. n corresponds number theoretically to the dimension for extension of rationals. This makes possible a hierarchy of quantum coherence length coming as n/6-multiples of the ordinary Compton length. Quantum coherence in long length scales is the most important implication and the coherence of living matter would be due to quantum coherence at magnetic body - distinguishing between TGD and Maxwellian and QFT view about classical fields. There is considerable evidence for the existence of h_eff hierarchy from various anomalies, in particular from those in living matter.
5. Also now the challenge is testing of this hypothesis in macroscopic length scales: we cannot directly access short scales. The idea is simple: measure ratios of p-adic mass scales. They do not depend on CP₂ scale nor on the value of n. The ratios of dark quantum scales - say dark Compton lengths - are typically given by the ratio n₁/n₂ of integers involved.

   This allows precise tests by measuring mass and length scale ratios rather than masses and length scales. For instance, the possibility of scaled variants of hadron physics and electroweak physics allow to test the hypothesis. There are indeed indications for scaled up variants of mesons with mass scale differing by a factor 512 from that for ordinary hadrons. The Compton lengths would be same as for ordinary hadrons for n/6=512: the dark Compton scale for p-adically scaled up meson be same as ordinary Compton length making possible resonant coupling. If the valence electron of atom is dark, its Bohr radius is scaled up by (n/6)²: these states might be misinterpreted as Rydberg states.

Concerning disretization of space-time TGD allows different view.

1. Physics as number theory vision predicts that hierarchy of extensions of rationals defines evolutionary hierarchy. A generalization of real numbers to adeles labelled by extensions of rationals is assumed. For given extension of rationals adeles form a book like structure having as pages real numbers and extensions of p-adic number fields induced by extension of rationals. The pages are glued together along the back of the book consisting of points in given extension of rationals common to reals and extensions of all p-adic number fields. This hierarchy corresponds to evolutionary hierarchy and the dimension n of extension has identification as effective Planck constant h_eff.
2. Space-time itself becomes a book-like structure. Real space-time surfaces are replaced with adelic surfaces, which containing real sheet and p-adic sheets glued together along the back of a book consisting of points with imbedding space coordinates in given extension of rationals. The points of space-time surface with coordinates in given extension of rationals form a discrete cognitive representaton, which is unique and improves with the dimension of extension so that at the limit of algebraic numbers it is dense set of space-time surface.
3. I call this discretization identifiable as intersection of sensory world (reals) and cognitive worlds as cognitive representation. The discretizaton reflects the limitations of cognition which must always discretize. In M⁸ picture space-time surface are "roots" of octonionic polynomials and the polynomial defines the extension of rationals via its roots. At M⁸ level there are also essentially unique imbedding space coordinates making discretization unique.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

What next in TGD?

Last night I was thinking about possible future project in TGD. The construction of scattering amplitudes has been the dream impossible that has driven me for decades. Maybe the understanding of fermionic M⁸-H duality provides the needed additional conceptual tools.

1. M⁸ picture looks simple. Space-time surfaces in M⁸ can be constructed from real polynomials with real (rational) coefficients, actually knowledge of their roots is enough. Discrete data - roots of the polynomial!- determines space-time surface as associative or co-associative region! Besides this one must pose additional condition selecting 2-D string world sheets and 3-D light-like surfaces as orbits of partonic 2-surfaces. These would define strong form of holography (SH) allowing to map space-time surfaces in M⁸ to M⁴×CP₂.
2. Could SH generalize to the level of scattering amplitudes expressible in terms of n-point functions of CFT?! Could the n points correspond to the roots of the polynomial defining space-time region!

   Algebraic continuation to quaternion valued scattering amplitudes analogous to that giving space-time sheets from the data coded SH should be the key idea. Their moduli squared are real - this led to the emergence of Minkowski metric for complexified octonions/quaternions) would give the real scattering rates: this is enough! This would mean a number theoretic generalization of quantum theory.

3. One can start from complex numbers and string world sheets/partonic 2-surfaces. Conformal field theories (CFTs) in 2-D play fundamental role in the construction of scattering string theories and in modelling 2-D statistical systems. In TGD 2-D surfaces (2-D at least metrically) code for information about space-time surface by strong holography (SH) .

   Are CFTs at partonic 2-surfaces and string world sheets the basic building bricks? Could 2-D conformal invariance dictate the data needed to construct the scattering amplitudes for given space-time region defined by causal diamond (CD) taking the role of sphere S² in CFTs. Could the generalization for metrically 2-D light-like 3-surfaces be needed at the level of "world of classical worlds" (WCW) when states are superpositions of space-time surfaces, preferred extremals?

The challenge is to develop a concrete number theoretic hierarchy for scattering amplitudes: R→C→Q→O - actually their complexifications.

1. In the case of fermions one can start from 1-D data at light-like boundaries LB of string world sheets at light-like orbits of partonic 2-surfaces. Fermionic propagators assignable to LB would be coded by 2-D Minkowskian QFT in manner analogous to that in twistor Grassmann approach. n-point vertices would be expressible in terms of Euclidian n-point functions for partonic 2-surfaces: the latter element would be new as compared to QFTs since point-like vertex is replaced with partonic 2-surface.
2. The fusion (product?) of these Minkowskian and Euclidian CFT entities corresponding to different realization of complex numbers as sub-field of quaternions would give rise to 4-D quaternionic valued scattering amplitudes for given space-time sheet. Most importantly: there moduli squared are real! A generalization of quantum theory (CFT) from complex numbers to quaternions (quaternionic "CFT").
3. What about several space-time sheets? Could one allow fusion of different quaternionic scattering amplitudes corresponding to different quaternionic sub-spaces of complexified octonions to get octonion-valued non-associative scattering amplitudes. Again scattering rates would be real. A further generalization of quantum theory?

There is also the challenge to relate M⁸- and H-pictures at the level of WCW. The formulation of physics in terms of WCW geometry leads to the hypothesis that WCW Kähler geometry is determined by Kähler function identified as the 4-D action resulting by dimensional reduction of 6-D surfaces in the product of twistor spaces of M⁴ and CP₂ to twistor bundles having S² as fiber and space-time surface X⁴⊂ H as base. The 6-D Kähler action reduces to the sum of 4-D Kähler action and volume term having interpretation in terms of cosmological constant.

The question is whether the Kähler function - an essentially geometric notion - can have a counterpart at the level of M⁸.

1. SH suggests that the Kähler function identified in the proposed manner can be expressed by using 2-D data or at least metrically 2-D data (light-like partonic orbits and light-like boundaries of CD). Note that each WCW would correspond to a particular CD.
2. Since 2-D conformal symmetry is involved, one expects also modular invariance meaning that WCW Kähler function is modular invariant, so that they have the same value for X⁴⊂ H for which partonic 2-surfaces have induced metric in the same conformal equivalence class.
3. Also the analogs of Kac-Moody type symmetries would be realized as symmetries of Kähler function. The algebra of super-symplectic symmetries of the light-cone boundary can be regarded as an analog of Kac-Moody algebra. Light-cone boundary has topology S²× R₊, where R₊ corresponds to radial light-like ray parameterized by radial light-like coordinate r. Super symplectic transformations of S²× CP₂ depend on the light-like radial coordinate r, which is analogous to the complex coordinate z for he Kac-Moody algebras.

   The infinitesimal super-symplectic transformations form algebra SSA with generators proportional to powers rⁿ . The Kac-Moody invariance for physical states generalizes to a hierarchy of similar invariances. There is infinite fractal hierarchy of sub-algebras SSA_n⊂ SSA with conformal weights coming as n-multiples of those for SSA. For physical states SSA_n and [SSA_n,SSA] would act as gauge symmetries. They would leave invariant also Kähler function in the sector WCW_n defined by n. This would define a hierarchy of sub- WCWs of the WCW assignable to given CD.

   The sector WCW_n could correspond to extensions of rationals with dimension n, and one would have inclusion hierarchies consisting of sequences of n_i with n_i dividing n_i+1. These inclusion hierarchies would naturally correspond to those for hyper-finite factors of type II₁.

   See the article Fermionic variant of M⁸-H duality or the chapter ZEO and matrices.

   For a summary of earlier postings see Latest progress in TGD.

   Articles and other material related to TGD.

M8-H duality for fermions

M⁸-H duality in bosonic sector is rather well understood but the situation is different in the fermionic sector. The basic guideline is that also fermionic dynamics should be algebraic and number theoretical.

1. Spinors should be octonionic. I have already earlier considered their possible physical interpretation.
2. Dirac equation as linear partial differential equation should be replaced with a linear algebraic equation for octonionic spinors which are complexified octonions. The momentum space variant of the ordinary Dirac equation is an algebrac equation and the proposal is obvious: PΨ=0, where P is the octonionic continuation of the polynomial defining the space-time surface and multiplication is in octonionic sense. The masslessness condition restricts the solutions to light-like 3-surfaces m_klP^kP^l=0 in Minkowskian sector analogous to mass shells in momentum space - just as in the case of ordinary massless Dirac equation. P(o) rather than octonionic coordinate o would define momentum. These mass shells should be mapped to light-like partonic orbits in H.
3. This picture leads to the earlier phenomenological picture about induced spinors in H. Twistor Grassmann approach suggests the localization of the induced spinor fields at light-like partonic orbits in H. If the induced spinor field allows a continuation from 3-D partonic orbits to the interior of X⁴, it would serve as a counterpart of virtual particle in accordance with quantum field theoretical picture.

   Addition: A really pleasant surprise that came this morning-9.7.2020 - sounds melodramatic but I do not want to forget it - it could have come more than decade ago but did not. The octonionic inner product for complexified octonionic 8-momenta with conjugation with respect to commuting imaginary unit i gives 8-D Minkowski norm squared. Same about quaternonic norm for complexified quaternionic momenta. Minkowski space with signature of (1,-,-1,-1) for metric follows from number theory alone! This conforms with the very idea of M⁸-H duality that geometry and number theory are dual in physics. Already this single finding makes M⁸-duality a "must".

   See the article Fermionic variant of M⁸-H duality or the chapter Does M⁸ duality reduce classical TGD to octonionic algebraic geometry?: Part III.

   For a summary of earlier postings see Latest progress in TGD.

   Articles and other material related to TGD.