Problem: The Landauer limit L(T) = kTln2 (k is the Boltzmann constant) means the energy corresponding to erasing a single bit of information. Thus, one can define t(T)=kTln2/ћ (ћ is the Planck constant). What might be the physical meaning of the so-defined function t(T) or that of its reverse function T(t)? Furthermore, t can be referred as f=1/t to the frequency of some wave having a certain physical sense.
Thesis: One possible meaning fundamental explicates an implicit mechanic-thermodynamic interpretation of quantum mechanics, and therefore the link between the irreversibility of time (involved by the Landauer limit for thermodynamics) and the reversibility of time (in turn linked to the reversible time in mechanics by the Plank constant).
A series of arguments in favor of the thesis:
1. The Planck constant means the correspondence of a dimensionless physical quantity to the quantity of action. That dimensionless physical quantity can be interpreted as entropy.
2. The Landauer limit means energy per a bit of information (entropy).3, Action is a function of energy and time and thus, time as temperature can be interpreted as a relation (e.g. ratio) of energy and entropy.
4. Then, the dependence of time and temperature at both Landauer limit and Planck constant for both time and temperature are interpreted only by energy and entropy as above implies both time and temperature to be referred to the same state of affairs from two different viewpoints: (quantum) mechanics for time, and thermodynamics for temperature.
5. That idea (4) implies that energy and entropy in turn can be seen as the same quantity from two different viewpoints, and time and temperature represent the difference between the latter two viewpoints from the former two viewpoints accordingly of (quantum) mechanics and thermodynamics.
6. That quantity described whether as time or as temperature means intuitively the transition "inside - outside", or a little more exact: the transition "inside of a physical system" to "outside of the same system". That transition in (quantum) mechanics is called motion and mathematically described as a function of time. It means that system as a single whole conventionally called "particle", which is moving from the past position ("inside") to the future position ("outside") right now. That transition seems not to be called in thermodynamics (at least, I cannot guess how), but it means the transition from considering the system as consisting of many, many elements (“inside”) to a thermodynamic whole (“outside”). Thus, time and temperature might be distinguished as the opposite directions of the same relation, namely time in the direction “outside - inside” (t=S/E, S may mean both action and entropy for the Planck constant), and temperature in “inside – outside” (t=E/S, S again both).
7. Consequently, frequency as the quantity reciprocal to time (as the period of a certain wave) can be seen as an equivalent of temperature. Indeed, the physical process of emission equates directly the temperature of the emitter to the frequency of the emitted. That process represents, in fact, wave-particle duality embodied physically as a kind of “particle-wave physical transformation” for the mechanical motion of a thermodynamic ensemble of particles is transformed directly into, and thus equated to a wave of light.
8. Quantum mechanics was forced by the Planck constant to unite the viewpoints of mechanics and thermodynamics. That unified viewpoint can be understood intuitively so. The quantum leap as the (quantum) mechanical motion as above can be seen as the thermodynamic transition “outside – inside” representable by a quantity reciprocal to temperature. Indeed, mechanical motion being always from the past to the present and future corresponds intuitively right to the transition “outside – inside” for we and any mechanical motion are always outside of the past just in the present.
9. Furthermore, quantum mechanics managed to represent the quantum leap by a hypothetical thermodynamic ensemble of all possible state of the leaping system right by a Gibbsian manner. Even more, Feynman reinterpreted that ensemble as a Boltzmannian ensemble of trajectories though only possible, but as a matter of ontology rather than as that of the corresponding mathematical formalism. That corresponding hypothetical thermodynamic ensemble can be thought as a special kind of wave, the medium of which consists of the elements of that ensemble, and its state by the states of those elements. That wave propagates inherently in phase space, in which a propagating wave can be associated to any evolving thermodynamic system represented in it.
10. The standard representation of quantum mechanics by the separable complex Hilbert space is equivalent to that phase-space representation under condition of the Planck constant determining the discrete structure of the corresponding phase space.
11. Thus, the fundamental postulate of wave-particle duality, implying by the way both quantum correlations and absence of hidden variables in quantum mechanics, can be equivalently reformulated to mechanic-thermodynamic duality, which can explain the dependence of time (for mechanics) and temperature (for thermodynamics) at both Landauer limit and Planck constant.
12. That dependence as well as “mechanic-thermodynamic duality” can be embedded naturally in an otology of (quantum) information as described below:
Any past moment is erased irreversibly and physically. However, all information in it is transmitted into the present moment and transformed equivalently in the quantity of action by the Planck constant. This recollects Descartes’s concept of time recreating the world again and again in any moment. That conserved information transferred into the present moment decreases its entropy and thus increases its temperature. One needs an equivalent influx of energy in the framework of the second law of thermodynamics requiring for entropy to decrease. That influx of energy will seem external and “dark” in origin for it has not any internal source. Its real origin is all information erased into the past and transferred into the present. That transition is leap-like. The information transfer is realized by quantum correlations and refers to irreversible time. It does not obey the light limit, but is represented onto the “screen of the present” obeying that limit and reversible time as gravity according to the theory of general relativity.
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