Could one apply the thermodynamical approach of England in TGD framework?
It turns out possible to gain amazing additional insights about TGD inspired view of life and consciousness by generalizing England's approach discussed in previous posting. Several puzzling co-incidences find an explanation in the thermodynamical framework and the vision about solar system as a living quantum coherent entity gains additional support.
- The situation considered in England's approach is a system - say biomolecule - in heat bath so that energy is not conserved due the transfer of energy between reactants and heat bath.
- The basic equation is equilibrium condition for the reaction i→ f and its time reversal f*→ i*. The initial and final state can be almost anything allowing thermodynamical treatment: states of biomolecule or even gene and its mutation. The ratio of the rates for the reaction and its time reversal is given by the ratio of the Boltzmann weights in thermal equilibrium:
R(i→ f)/R(f*→ i*)= R , R= e-(Ei-Ef)/T .
Ei and Ef denote the energies of initial and final state. This formula is claimed to hold true even in non-equilibrium thermodynamics. It is important that the ratio of the rates does not depend at all on various coupling constant parameters. The equilibrium condition must be modified if initial and final states are fermions but it is assumed that states can be described as bosons. Note that in heat bath even fermion number need not be conserved.
- If the energy eigenstates are degenerate, the ratio R of Boltzman factors must be modified to include the ratio of state degeneracies
R→ (D(Ei)/D(Ef) × e-(Ei-Ef)/T .
This generalization is essential in the sequel.
One can imagine two possible reasons for the presence of exponentially large factors compensating Boltzmann weights D(Ei). The first reason is that for heff=n× h the presence of n-fold degeneracy due to the n-fold covering of space-time surface reducing to 1-fold covering at its ends at the ends of CD is essential. Second possible reason is that the basic object are magnetic flux tubes modellable as strings with exponentially increasing density of states. These mechanisms could quite well be one and same.
- Since magnetic flux tubes are key entities in TGD inspired quantum biology, stringy dynamics suggests itself strongly. The situation thus differs dramatically from the standard biochemical situation because of the presence of dark matter at magnetic flux tubes to which one can assign fermion carrying strings connecting partonic 2-surfaces defining correlates for particles in very general sense.
- The key aspect of stringy dynamics is Hagedorn temperature. Slightly below Hagedorn temperature the density of states factor, which increases exponentially, compensates for the Boltzmann factor. Hagedorn temperature is given by
THag = (61/2/2π) × (1/α')1/2 ,
where α' is string tension. In superstring models the value of string tension is huge but in TGD framework the situation is different. As a matter fact, the temperature can be rather small and even in the range of physiological temperatures.
- What makes THag so special is that in the equilibrium condition reaction and its reversal can have nearly the same rates. This could have profound consequences for life and even more - make it possible.
In ZEO based quantum measurement theory and theory of consciousness time reversal indeed plays key role: self dies in state function reduction to the opposite boundary of CD and experiences re-incarnation as a time-reversed self. This process is essential element of memory, intentional action, and also remote metabolism, which all rely on negative energy signals travelling to geometric past assignable to time reversed sub-selves (mental images). The above formula suggests that intelligent life emerges near THag, where the time reversed selves are generated with high rate so that system remembers and pre-cognizes geometric future as it sleeps so that memory planned action are possible.
- String tension cannot be determined by Planck length as in string models if it is to be important in biology. This is indeed the case in TGD based quantum gravity. The gravitational interaction between partonic 2-surfaces is mediated by fermionic strings connecting them. If string tension were determined by Planck length, only gravitational bound states of size of order Planck length would be possible. The solution of the problem is that the string tension for gravitational flux tubes behaves like 1/heff2.
In TGD framework string tension can be identified as an effective parameter in the expression of Kähler action as stringy action for preferred extremal strongly suggested by strong form of holography (SH) allowing the description of the situation in terms of fermionic strings and partonic 2-surfaces or in terms of interiors of space-time surfaces and Kähler action. 1/heff2 dependence can be derived from strong form of holography assuming electric-magnetic duality for Kähler form, and using the fact that the monopoles associated with the ends have same magnetic and electric charges.
- The discussion of the analog of Hawking radiation in TGD framework led to an amazing prediction: the TGD counterpart of Hawking
temperature turns out to be in the case of proton very near to the physiological temperature if the big mass is solar mass (see this). This suggests that the entire solar system should be regarded as quantum coherent living system. This is also suggested by the general vision about EEG. Could Hawking temperature be near to the Hagedorn temperature but below it?
- In ZEO the notion of heat bath requires that one considers reactants as subsystems. The basic mathematical entity is the density matrix obtained by tracing over entanglement with environment. The assumption that dark matter is in thermal equilibrium with ordinary matter can be made but is not absolutely crucial. The reactions transforming visible photons to dark photons should take care of the equilibrium. One could even assume that the description applies even in case of the negentropic entanglement since thermodynamical entropy is different from entanglement entropy negative for negentropic entanglement.
- In TGD inspired quantum biology one identifies the gravitational Planck constant introduced by Nottale with heff=n× h. The idea is simple: as the strength of gravitational interaction becomes so strong that perturbation series fails to converge, a phase transition increasing the Planck constant takes place. hgr=GMm/v0= heff=n× h implies that v0/c<1 becomes the parameter defining the perturbative expansion. hgr is assigned with the flux tubes mediating gravitational interaction and one can say that gravitons propagate along them.
Note that this assumption makes sense for any interaction - say in the case of Coulomb interaction in heavy atoms: this assumption is indeed made in the model of leptohadrons (see this) predicting particles colored excitations of leptons lighter the weak bosons: this leads to a contradiction with the decay widths of weak bosons unless the colored leptons are dark. They would be generated in the heavy ion collisions when the situation is critical for overcoming the Coulomb wall.
The cyclotron energy spectrum of dark particles at magnetic flux tubes is proportional to hgr/m does not depend on particle mass being thus universal. In living matter cyclotron energies are assumed to be in the energy range of bio-photons and thus includes visible and UV energies and this gives a constraint on hgr if one makes reasonable assumption about strengths of the magnetic fields at the flux tubes (see this). Bio-photons are assumed to be produced in the transformation of dark photons to ordinary photons. Also (gravitational) Compton length is independent on particle mass being equal to Lgr=GM/v0: this is crucial for macrosopic quantum coherence at gravitational flux tubes.
- The basic idea is that Hawking radiation in TGD sense is associated with all magnetic flux tubes mediating gravitational
interaction between large mass M, say Sun, and small mass m of say elementary particle. How large m can be, must be left open.
This leads to a generalization of Hawking temperature (see this) assumed to make sense for all astrophysical objects at the flux tubes connecting them to external masses:
TGR=hbar (GM/RS2 2π) = hbar/(8 π GM).
For Sun with Schwartschild radius rS=2GM=3 km one has TGR= 3.2× 10-11 eV.
Planck constant is replaced with hgr=GMm/v0= heff=n× h in the defining formula for Hawking temperature. Since Hawking temperature is proportional to the surface gravity of blackhole, one must replace surface gravity with that at the surface of the astrophysical object with mass M so that radius RS=2GM of the blackhole is replaced with the actual radius R of the astrophysical object in question. This gives
THaw= (m/8 π v0) ×(RS/R)2 .
The amazing outcome is that for proton the estimate for the resulting temperature for M the solar mass, is 300 K (27 C), somewhat below the room temperature crucial for life!
Could Hagedorn temperature correspond to the highest temperature in which life is possible - something like 313 K (40 C)? Could it be that the critical range of temperatures for life is defined by the interval [THaw,THag]? This would require that THaw is somewhat smaller THag. Note that Hawking temperature contains the velocity parameter v0 as a control parameter so that Hawking temperature could be controllable. Of course, also THaw=THag can be considered. In this case the temperature of environment would be different from that of dark matter at flux tubes.
- The condition THaw≤ THag allows to pose an upper bound on the value of the effective string tension
(α')-1/2≥ (m/4×61/2v0) × (RS/R) .
For a summary of earlier postings see Links to the latest progress in TGD.