About information thermodynamics and Maxwell's demon in TGD

Information thermodynamics includes information as an analogy of negative entropy. The associated temperature can be assumed to be the same as the usual temperature but this need not be the case. Information thermodynamics leads to a generalization of the second law taking into accout the presence of quantum information. Whether this information corresponds to conscious information, is left open.

Sciencedaily article (see this) tells about the article of Minagawa et al (see this) published in Nature, in which a more rigorous proof than before is represented for the statement that that the second law of thermodynamics is also valid in quantum theory with information included. The presence of information makes possible an apparent local violation of the second law in the sense that the system can do more work than the Carnot law allows. However, Maxwell’s demon requires metabolic energy to function and it loses it when demonizing. This energy must be compensated and the conclusion is that the second law of thermodynamics is valid.

I have discussed Maxwell’s demon earlier in (see this). In the following information dynamics and Maxwell’s demon are discussed in light of the recent results related to TGD (see this, this, this, this and this). The discussion relies on the geometric vision of physics involving holography= holomorphy principle and the number theoretic vision of physics involving p-adic number fields and adeles as mathematical correlates for cognition and the evolutionary hierarchy of extensions of rationals. Negentropy Maximization Principle (NMP) for cognitive information replacing second law and implying it. Zero energy ontology (ZEO) provides the new quantum ontology.

1. Some basic ideas of TGD relevant to Maxwell’s demon

Consider first briefly the geometric vision.

1. In the classical TGD, holography= holomorphy vision is a new element not present in the earlier discussion of Maxwell’s demon discussed in (see this). Space time surfaces as roots of analytic function pairs f=(f₁,f₂): H=M⁴× CP₂ → C² with Taylor coefficients in some extension E of rationals provide exact solution of field equations by reducing them from partial differential equations to algebraic equations. (see this). Polynomial solutions are obtained as an important special case.

Also the roots g(f)=(0,0), g: C²→ C² define space-time surfaces. The hierarchy obtained as composites of maps g gives rise to a hierarchy of field bodies. This hierarchy of maps of maps of … is analogous to an abstraction hierarchy. For g(0,0)=(0,0) the lowest level (f₁,f₂)=(0.0) belongs to the hierarchy (see this).

The algebraic complexity of the surfaces increases with the number of composites g_i and their inherent complexity measured by degrees with respect to the 3 complex coordinates of H. When g reduces to map C→ C one has ordinary polynomials and one can assign Galois group and ramified primes to it. The “world of classical worlds” (WCW) decomposes to a union of sub-WCWs with f₂ fixed.

Field/field body (see this and this) serves as a carrier phases of the ordinary matter with non-standard value of effective Planck constant h_eff≥ h making it a quantum coherent system in arbitrarily long scales. The proposal is that h_eff corresponds to the dimension of an algebraic extension of rationals as the order of the Galois group. In the TGD inspired quantum biology, the temperature of the field body is not necessarily the same as the temperature of the biological body, but could be lower and would gradually increase with aging (see this) so that the field body would gradually lose control.

Zero energy ontology (ZEO) (see this and this), forced by the holography, in turn forced by the general coordinate invariance, makes classical physics an exact part of quantum TGD.

Holography involves space-time surfaces identified as classical Bohr orbits for particles of 3-surfaces as generalization of point-like particles and localized inside CD. These Bohr orbits are slightly non-deterministic already for 2-dimensional minimal surfaces. This forces the replacement of quantum states with superpositions of Bohr orbits and brings in new degrees of freedom related to classical non-determinism. These degrees of freedom are essential for understanding cognition. The identification of quantum states as superpositions of Bohr orbits allows us to solve the basic problem of quantum measurement theory. In ZEO both arrows of time are possible in all scales.

The hierarchy of causal diamonds CD= cd× CP₂, where cd is causal diamond of M⁴ is the geometric correlate of ZEO. CD has interpretation as a 4-D perceptive field of a conscious entity associated with the quantum superpositions of 4-surfaces inside CD. CDs form a scale hierarchy (see this).

In the TGD framework, information means potentially conscious information and involves TGD view of memory based on the classical non-determinism (see this). Number theoretic vision is complementary to the geometric view about TGD.

1. p-Adicization and adelization of cognition. Negentropy and its energy equivalent. Negentropy could correspond to information in information thermodynamics. The production of negentropy requires metabolic energy. Cognitive negentropy as the sum of p-adic negentropies increases, but so does real entropy. This fits (see this and this) with Jeremy England’s observations (see this). Classical non-determinism could correspond to p-adic non-determinism.
2. In the TGD framework, one can speak of the Galois group in two different senses (see this, this and this). TGD predicts a 4-D variant of Galois group mapping different regions of the space-time surfaces identified in holography= holomorphy vision as roots (f₁,f₂)=(0,0) for function pairs H=M⁴× CP₂→ C² analytic with respect to Hamilton-Jacobi coordinates generalizing complex coordinates (see this). The 4-D Galois group is realized as analytic flows analogous to braidings mapping the roots as space-time regions to each other.

The second Galois group is associated with dynamical complex analytic symmetries g: C→ C: (f₁,f₂)→ (g((f₁,f₂)). One can talk of number theoretic/topological n-ary digits for n-sheeted space-time surfaces. Binary digits (n is prime) are in a well-defined sense fundamental. In this case, the Galois group relates to each other disjoint space-time surfaces. When g reduces to map C→ C, one can assign to it an ordinary Galois group relating to each other the disjoint roots of g(f).

These two Galois groups commute and the latter Galois group relates to the first Galois group in the same way as the Galois group of an extension of rationals to the Galois group of complex rations generated by complex conjugation.

Prime polynomials are polynomials, which cannot be expressed as composites of two polynomials. If the degree of the polynomials f_i are prime with respect to the 3 complex coordinates of H, this is the case. Same applies to the polynomials (g₁,g₂): C²→ C². These kinds of surfaces are analogs to elementary particles. Finite group is prime/simple, if it does not allow normal subgroups. Prime Galois groups could be associated with g: C→ C are expected to be in a special role physically.

Does the notion of Galois group generalize to the general case g=(g₁,g₂)? Also in this case the roots of g(f) are disjoint space-time surfaces representing pairs of algebraic numbers (f₁,f₂)=(r_i,1,r_i,2). Is it possible to assign to the roots the analog of the Galois group? This group should act as a group of automorphisms of some algebraic structure. This structure cannot be a field but algebra structure is enough. These arithmetic operations would be component-wise sum (a,b)+(c+d)=(a+c,b+d) and componentwise multiplication (a,b)*(c,d)= (ac,bd). The basic algebra would correspond to the points of (x,y)in E² or rationals and the extension would be generated by the pairs (f₁,f₂)= (r_i,1,r_i,2). This structure has an automorphism group and would serve as a Galois group. The dimension of the extension of E² could define the value of the effective Planck constant.

Also the notion of discriminant can be generalized to a pair (D₁,D₂) of discriminants using the component-wise product for the differences of root pairs. Could D_i be decomposed to a product of powers of algebraic primes of the extension of E²?

In (see this) the idea that the space-time surfaces can be regarded as numbers was discussed. For a given g, one can indeed construct polynomials having any for algebraic numbers in the extension F of E defined by g. g itself can be represented in terms of its n roots r_i=(r_i,1,r_i,2), i=1,n represented as space-time surfaces as a product ∏_i(f₁-r_i,1,f₂-r_i,2) of pairs of monomials. One can generalize this construction by replacing the pairs (r_i,1,r_i,2) with any pair of algebraic numbers in F. Therefore all algebraic numbers in F can be represented as space-time surfaces. Also the sets formed by numbers in F can be represented as unions of the corresponding space-time surfaces.

This picture also leads to a vision about physics laws as being analogous to laws of logic and time evolution of the physical system as analogous to a proof of theorem (see this). Different meta levels defined by the maps g: C²→ C² are analogous to hierarchy of statements about statements in mathematics. This applies also to the more general maps. The interpretation is that the surfaces at the higher levels of the meta hierarchy represent statements about the surfaces (f₁,f₂)=(0,0) at the lowest level of the hierarchy.

The creation of algebraic complexity, i.e. increasing the value of h_eff, requires metabolic energy. The value of h_eff tends to spontaneously decrease, which gives rise to a dissipation as a competing effect and one should understand how these two tendencies relate to each other.

Negentropy Maximization principle (see this) states that the algebraic complexity measured by the sum of p-adic negentropies increases in a statistical sense during the number theoretic evolution and NMP states this fact. Also the ordinary entropy increases as a result of the increase of the negentropy and this conforms (see this) with the view of Jeremy England (see this. The maximum value for h_eff partially characterizes algebraic complexity. Also Galois groups, the degree(s) of the polynomials defining the space-time surface, and ramified primes (when they are defined) characterize the complexity. Galois groups generalize the group Z₂ assigned with condensed matter fermions in topological quantum computation (see this).

2. The TGD view of Maxwell’s demon

The basic vision is as follows.

1. A field body can act as a Maxwell’s demon. As a carrier of potentially conscious information realized as memories about previous SSFRs (see this), the field body can transfer negentropy, basically energy, to lower levels and seemingly help to break the second law. As a result, an additional term from information to the basic equation expressing the second law. The h_eff for the field body however decreases in the process.

Also the classical fields associated with the field body can do work on the biological body and the classical gravitational/electric fields are characterized by gravitational/electric Planck constants (see this and (see this.

In presence of the field bodies, perhaps assignable to the hierarchy of abstractions defined by the maps g, more work can be obtained from the system than the second law would otherwise allow. This is only apparent because the h_eff of the field body decreases so that the algebraic complexity measuring the level of consciousness decreases. Therefore h_eff must be increased to its original value. Metabolic energy is needed for this purpose. Consider now the TGD view of Maxwell’s demon as a conscious entity in light of the recent results not yet taken into account in the discussion of (see this).

1. Consider first the notion of a cognitive measurement. The Galois group for g₁(g₂(…), g_i: C→ C is characterized by a hierarchy of coset groups defined by the hierarchy of its normal subgroups. The irreps of the Galois group can be decomposed to tensor products of the coset groups and the entanglement between the coset groups can be measured in cognition SFRs.

In particular, the hierarchy formed by functional composition of polynomials g: C→ C is accompanied by a hierarchy of extensions of extensions … of E and the hierarchy of normal subgroups for the Galois groups.

Prime groups are simple groups having no normal groups and prime polynomials do not allow functional decomposition P(Q). If the degree of the polynomial is prime this is the case but this is not a necessary condition. Space-time surfaces of this kind are fundamental objects and the polynomial in question would have prime degree with respect to the 3 complex coordinates of H. The composites formed of maps g and of these fundamental function pairs f would define cognitive representations of the surface defined by f as kind of statements about statements. An interesting question is whether these surfaces could correspond to elementary particles. The primes orders of polynomials involved do not probably correspond to p-adic primes assigned in p-adic mass calculations to elementary particles and identified as ramified primes (see this).

What happens when the value of h_eff descreases? Does it decrease only apparently? If the highest levels in the tensor product of irreps of the coset groups become singlets, these levels effectively drop away. Or could the space-time surface itself become simpler as some maps g in the composite g₁(g₂(…) become identity maps? Field body would lose some of its levels. Indeed, in the TGD based view of biocatalysis, the temporary reduction of h_eff is essential for the biocatalysis (see this).

3. How could the time reversal in BSFR relate to the second law?

Time reversal in BSFRs represents new physics. What are the implications?

1. Entanglement entropy and p-adic negentropies are associated with a pair of systems. Thermodynamic entropy is associated with an ensemble. A single BSFR cannot reduce ensemble entropy. It reduces entanglement entropy. Quantum entanglement with the environment is the cause of BSFR. Ensemble entropy, however, corresponds to entanglement entropy after a quantum measurement when quantum entanglement has disappeared.
2. In the ZEO based theory of consciousness, falling asleep (or biological death) corresponds to a BSFR and waking up to another BSFR. Sleeping restores resources and heals. What does this mean? Does the field body gain new metabolic energy resources that appear after the second BSFR? Does the BSFR violate the second law or does the energy needed come from outside the process as metabolic energy?

During sleep, metabolic energy is not used for normal bodily purposes such as movement and sensation. It could go to the field body to restore the values of h_eff to their original values. For example, protons are transferred back to dark protons by the Pollack effect. That would require metabolic energy. This would be visible as the presence of an unknown energy sink during sleep. Metabolic energy consumption would not be reduced.

During sleep, the second law applies in the opposite direction of time. This would only allow for an apparent violation of the second law.

BSFR would correspond to the loss of entanglement between the system and the environment, which for an ensemble means an increase in ordinary entropy. The produce thermodynamic entropy would be equal to as entanglement entropy transformed into ensemble entropy in BSFRs. Thermodynamic entropy for the ensemble would increase for both arrows of time in BSFRs.

The subsystems as smaller space-time sheets topologically condensed on the space-time sheet of the system can form a thermodynamic ensemble. The entropy of this ensemble increases in both directions of time. If it is possible to observe its increase occurring in the opposite direction of time, it would look like a decrease in entropy in the standard time direction and give rise to apparent violation of the second law. In the Pollack effect this kind of effect is observed for the negatively charged exclusion zones at EZs throw impurities out. This looks like time reversed diffusion and can be understood in the ZEO (see this).

In the case of SSFRs it is not meaningful to talk about an ensemble. The SSFRs cannot not therefore affect the ensemble entropy. The sequences of SSFRs define the analog of an adiabatic time evolution.

In biosystems the metabolic energy input makes it possible to maintain the h_eff distribution. At lower levels, entropy of the field body however increases gradually and eventually leads to death. Is it possible to maintain the metabolic energy feeds or are the metabolic energy currents gradually reduced during the evolution so that there is no metabolic energy feed increasing the complexity for subsystems anymore? This would be reduction of gradients leading to thermal equilibrium and loss of information. Could this be the TGD counterpart of heat death?

Could the pairs of BSFRs solve the problem by allowing a fresh start with the original arrow of geometric time with the field body having a pure state and low entropy after the second BSFR? Sleep begins and ends with a BSFR. Sleeping overnight indeed allows one to wake up full of energy which is dissipated during the day. Could BSFRs generate the necessary gradients needed to avoid heat death? Could BSFRs take care that the system’s form kind of flip flops exchanging metabolic energy by dissipating. See the article About information thermodynamics and Maxwell’s demon in TGD.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/03/about-information-thermodynamics-and.html