# Relating a System’s Hamiltonian to Its Entropy Production Using a Complex Time Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}/c, x

_{1}, x

_{2}, x

_{3}, where, as usual, c is the speed of light and i ≡ √ − 1, such that the squares of the unit vectors are −1, +1, +1, and +1, respectively, or simply “−+++”). However, the inverse metric for spacetime (real time and imaginary space: “+−−−”) is also valid, and we will show where each metric is physically appropriate.

**QGT**; Parker & Jeynes 2019 [6]), they have a similar description: both have an entropy determined by the Bekenstein–Hawking relation, a relation that is a necessary (holographic) consequence of the entropic Liouville theorem (see Parker and Jeynes 2021a [7]). Both are unitary entities in QGT, that is, there exist no simpler entities at that scale. Curiously, although both are maximum entropy entities (being unitary), the one has zero entropy production and the other has positive (non-zero) entropy production. However, even though the black hole necessarily grows, it still remains the same unitary entity. The entropy production of black holes was calculated previously (Parker and Jeynes 2021b [5]); here, we present an alternate derivation of the same result. It turns out that the matter radius of the alpha (and other nuclei) is readily calculated by QGT (Parker et al. 2022 [8]), thus it is helpful to show that these complex time methods are also valid for what might have been thought a trivial case.

## 2. Complex Time Descriptions for Reversibility and Irreversibility

_{B}and the diffusion coefficient D) and Schrödinger’s equation (describing the reversible evolution of the quantum wave function Ψ, and involving Planck’s constant h and the mass m of the particle, respectively); note that both equations are temporal in character, but we employ different symbols for their respective time variables, because Equation (1a) is essentially real in nature, whereas Equation (1b) is imaginary. Although different, using the diffusion coefficient definitions of Mita [9] and Nelson [10], the above two equations are related by the following:

_{B}is a reflection of an explicitly entropic version of the partition function, being composed of probabilities that are analogous to the modulus-squared of Schrödinger’s equation (see Parker and Jeynes 2021 [7], Equation (14d)), as is also described by Córdoba et al. 2013 [12] (see their Equations (29) and (30)).

_{B}), it is clear that an imaginary diffusion coefficient can also be described: γ = ik

_{B}/m. Of course, this begs the question of what determines whether the “diffusion coefficient” is real (describing an irreversible process) or imaginary (reversible).

**QGT**; Parker and Jeynes 2019 [6]).

_{cl}and the thermodynamic entropy S

_{th}across complex time z $\equiv -\tau +\mathrm{i}t$ (Equation (3)), which explicitly combines action and entropy. It is the unifying idea of complex time that allows the application of complex function theory across the complex temporal plane to coherently define the (dimensionless) analytic function $\mathcal{S}$ (see Equation (7) of [4]):

_{cl}and the thermodynamic entropy S

_{th}(see Equation (14) of [4]):

_{z}(where the subscript z indicates its relation to the complex temporal plane z) and Π is the real (irreversible) part of the complex entropy production Π

_{z}. Note that, where Velazquez et al. [4] use the term “entropic Hamiltonian”, we use instead the term “entropy production” as a synonym for “rate of entropy increase”.

_{z}and (complex) entropy production ${\mathsf{\Pi}}_{z}$ in any physical process must be understood from the trajectory across complex time taken by the physical phenomenon. That is, whether the process is reversible or irreversible (or a mixture) will be determined by how the real and imaginary components are combined at each point in time. We point out parenthetically that the representations we use seem ambiguous at this stage as to whether reversibility is indicated by the real or by the imaginary components. However, the usual convention is to regard the imaginary temporal t-axis as the reversible one, in the context of a complex temporal plane providing the comprehensive framework in which both reversible and irreversible processes can be consistently and completely described.

_{z}and ${\mathsf{\Pi}}_{z}$ are comparable to the expressions used in signal processing for analytic quantities (in particular, photons). QGT [6] shows how meromorphic functions are used to express information and how holomorphic functions express maximum entropy systems. John Toll [16] explicitly gives a rigorous proof that strict causality is logically equivalent to the existence of the “dispersion relations”, which are best known as practical constraints in signal processing, so that, in optics (for example), the refractive index has an imaginary component in the presence of absorption. However, as any particle can be represented as a wave, any scattering process must have a representation in terms of a “frequency distribution”, with the corresponding “group” and “phase” velocities. Toll has shown how the real and imaginary properties of the dispersion are mutually related via the Kramers–Kronig relations, using the properties of the Hilbert transform. Toll further points out that exactly the same formalism is applicable generally; not only to optics but also to (for example) high-energy particle scattering (citing the “excellent discussion” of the so-called “R-matrix” representation by Wigner [17]).

## 3. Fourier and Hilbert Transform Relations

_{0}, where t

_{0}is chosen as a convenient point in time to express the causality of the system) and is a physically realisable (square-integrable) function, then Cauchy’s theorem applies, and the Hamiltonian is holomorphic in the required (upper, as appropriate) half-plane, such that it obeys the dispersion relations. Following Toll [16] Equation (2.5), we can write the dispersion of the complex Hamiltonian (using the terms of Equation (8a) above) in terms of the component ω of the complex frequency $\widehat{\omega}$:

_{z}(see (Equation (8a)) are Hilbert transforms of each other (Equation (10)), they are indeed causal. The corollary is that Equation (10a) implies Equation (10b), and vice versa. The integration is performed parallel to the ω-axis and the analytical continuation into the upper half-plane exists.

_{t}and H

_{τ}, see Equations (8), and we shall see that Parseval’s Theorem, as applied to the respective Hilbert transform components, also provides additional useful insight into their physical properties). Thus, exploiting the mathematical properties associated with the process of analytical continuation, we may write two symmetrical pairs of expressions for how the complex entropy production function and the complex Hamiltonian function relate along the two conjugate frequency axes forming the complex frequency plane:

## 4. Application: The Alpha Particle

_{0}(with the subscript ‘0′ signifying the zero-frequency or d.c. value) are constants independent of any frequency variation. For the (unconditionally stable) alpha particle, we must also have ${\mathsf{\Pi}}_{0}$ = 0. In this case, the inverse Hilbert transform of ${\mathsf{\Pi}}_{\alpha}\left(\omega \right)$ (the second term associated with the RHS of Equation (27b)) is zero, so the Hamiltonian of the alpha is simply given by the constant H

_{0}(independent of frequency): H

_{0}≡ H

_{α}= p

^{2}/2m = −($\hslash $

^{2}/2m)∇

^{2}(see Equation (26a)).

## 5. Application: A Decaying Harmonic Oscillator

_{0}for the exponential decaying system is given by τ

_{0}= 1/$\alpha $, so that the appropriate ‘cut-off’ frequency for the integral (below which we can assume most of the entropy production is associated) is $\alpha $, such that we simply divide the integral by $\alpha $ (as indicated below) to determine the average entropy production. According to the usual conventions of measurement theory, we only consider the positive frequency components:

_{B}Tln2 of energy is associated with the erasure (or loss) of a bit of information, where a bit is also equivalent to a system degree of freedom (see also [8]). Thus, it is clear that, for a decaying oscillator, as its energy is dissipated, Equation (36) suggests that each of its degrees of freedom dissipates a quantity of energy $\alpha \hslash $. Clearly, when the oscillator is dissipationless and $\alpha $= 0, then, also according to Equation (36), none of its degrees of freedom are dissipating any energy.

## 6. Application: The Black Hole

_{BH}, which has a constant entropy production associated with the Hawking radiation (HR, see Parker and Jeynes [5] Equation (25)):

_{BH}within the Schwarzschild radius; yet, as is well known, any Hamiltonian can also be offset by a constant quantity (with there being no absolute value for the energy, see Caticha 2021 [20]), so that the value of the associated Hamiltonian as determined by Equation (27b) can be additionally offset by the energy lost to the Hawking radiation H

_{0}= M

_{BH}c

^{2}− H

_{HR}as required. Note that such an offset (by unity) is also seen in the Kramers–Kronig expression for the real part of the refractive index to obtain the correct Hilbert transform relationships (see, as an example, the unity offset in Equation (1.1) of Toll [16]). That is to say, we can now rewrite Equation (43) as follows:

_{HR}term on the RHS of Equation (44).

_{z}and ${\mathsf{\Pi}}_{z}$, which are normally degenerate, but which may become apparent at the extreme physical conditions of a black hole. Thus, Equations (22) suggest the following identities for the ‘trans-axial’ quantities, which may become apparent at the black hole event horizon where the metric is assumed to invert:

_{HR}Hawking radiation term in Equation (44). Actually, substituting the Hawking radiation Equation (37) into Equation (45b) now directly offers us the associated (integrated) energy Hamiltonian of the (very small) Hawking radiation:

_{P}and what we could call the “Planck entropy production” ${\mathsf{\Pi}}_{\mathrm{P}}$ intimately connecting the conventional and trans-axial Hamiltonian and entropy production quantities, respectively:

_{P}is the Planck time. The Planck entropy production quantity ${\mathsf{\Pi}}_{\mathrm{P}}$ is the same as the entropy production term given previously (Parker and Jeynes [5] Equation (31); the factor 2π comes from an ambiguity in the definition of wavenumber) in the context of the entropy production of spiral galaxies. Thus, we find that, associated with a black hole, there exists another very large entropy production term ${\mathsf{\Pi}}_{\mathrm{P}}$ (Equation (49)) that is 46 orders of magnitude larger than the term associated with the Hawking radiation ${\mathsf{\Pi}}_{HR}$ (Equation (37)); using the example of the Milky Way featuring a supermassive BH of mass 4.3 × 10

^{6}solar masses), and which can be understood as being related to the highly energetic processes seen occurring in the accretion disk surrounding a black hole.

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Parker, M.C.; Jeynes, C.
Relating a System’s Hamiltonian to Its Entropy Production Using a Complex Time Approach. *Entropy* **2023**, *25*, 629.
https://doi.org/10.3390/e25040629

**AMA Style**

Parker MC, Jeynes C.
Relating a System’s Hamiltonian to Its Entropy Production Using a Complex Time Approach. *Entropy*. 2023; 25(4):629.
https://doi.org/10.3390/e25040629

**Chicago/Turabian Style**

Parker, Michael C., and Chris Jeynes.
2023. "Relating a System’s Hamiltonian to Its Entropy Production Using a Complex Time Approach" *Entropy* 25, no. 4: 629.
https://doi.org/10.3390/e25040629