# On the Spectral Entropy of Thermodynamic Paths for Elementary Systems

## Abstract

**:**

## 1. Introduction

_{B}is the Boltzmann entropy constant. Because of energy exchanges between the system and surroundings, ρ necessarily fluctuates about an average. It can be shown that the ratio between the density standard deviation σ

_{ρ}and the average < ρ > is [3,4,5]:

_{T}is the isothermal compressibility:

^{23}particles/m

^{3}(i.e., p ≈ 400 Pascals at room temperature), then σ

_{ρ}/ < ρ > ≈ 3 × 10

^{-12}and σ

_{ρ}≈ 3 × 10

^{11}particles/m

^{3}. Three standard deviations of the number density (σ

_{ρ}) would correspond to c. 10

^{12}particles/m

^{3}. Repeated laboratory measurements of ρ for a 1 m

^{3}volume would manifest a narrow distribution about the average. More than 99% of the readings would fall in the range 10

^{23}+/− 10

^{12}particles/m

^{3}. A probability density plot based on the measurements would yield a near δ-function as in Panel B; higher density conditions only sharpen the function. If alternatively the ρ-time dependence were monitored, a recording as in Panel C would obtain. Here the particle density is shown to fluctuate about the average in a noisy fashion. A Fourier synthesis (or transform) would identify a zero-frequency (ω) component as the dominant one. The power spectrum in Panel D based on the Fourier analysis would evidence a single peak at ω = 0. Given the slight impact of energy exchanges between the system and surroundings, the amplitude is featureless and nearly zero for ω > 0.

**Figure 1.**Equilibrium Systems and Fluctuations. Panel A depicts a gas in equilibrium with its surroundings. Panel B shows the probability function that would obtain from repeated density measurements. Panel C illustrates the density behavior over time. Panel D shows the Fourier spectrum of the density behavior.

**Figure 2.**Systems and Variable Tuning. Panel A shows a system in which pressure and volume are tuned in parallel with work and heat exchanges. Panel B illustrates one of infinite possible pathways that connect the initial and final states. Panels C and D present alternative representations of the pathway. For simplicity, the system has been taken to be 1.00 mole of a monatomic ideal gas. The use of liter and atmosphere units follows the practice of classic thermodynamic texts [2,6,7].

## 2. Thermodynamic Pathways, Information, and Spectral Entropy

_{i},V

_{i}} sequence. In effect, ordered pairs of p

_{i}, V

_{i}enable the system to be stepped precisely along a chosen pathway. There is more than one input program which can accomplish the task. The K + 2 criterion for specifying state points allows other control variable pairs and sequences to be equally effective: {p

_{i},T

_{i}}, {ρ

_{i},V

_{i}}, and so forth.

_{rec}and heat received Q

_{rec}.

**Figure 3.**Pathways and Information. The upper portion schematically illustrates how a control variable sequence {p

_{i},V

_{i}} operates as a thermodynamic algorithm for pathway traversal: ordered pairs p

_{i}, V

_{i}enable the system to be stepped precisely through a succession of state points. Measurements via a transducer (triangle symbol) such as a McLeod gauge reduce uncertainty ? and trap information about the system, i.e., convert “?” to “yes” or “no”. The amount of information depends on the number and distribution of pathway state points.

_{i}} and {−log

_{2}prob

_{i}}, respectively. The expectation value of the surprisals quantifies the pathway information I

_{Y}in bits:

_{Y}is enhanced if the number of terms in Equation 7 is increased. This would be brought about by augmenting the pathway, or by extending the measurements at higher resolution—narrower Δp. I

_{Y}would be further enhanced if the probability terms proved equal (or nearly so) in value; this applies typically to pathways that evince complex structures. I

_{Y}is zero for certain quantities for certain pathways: isobaric, isothermal, and adiabatic pathways are absent in Y ↔ p, T, and S information, respectively. All closed systems pose zero information regarding the particle number: I

_{Y}

_{↔N}= 0.

_{Y}is an immediate by-product of information. There indeed exists S

_{Y}computable for every pathway state property, i.e., Y ↔ p, V, T, U, G, S, etc. Unlike information, S

_{Y}does not stem from yes/no queries and measurements with thermometers, pressure gauges, and so forth. Its significance arises instead because of the contact made with the algorithmic structure. Most notably, S

_{Y}quantifies the symmetry, or lack of it, imbedded in a pathway. Figure 4, for example, shows how S

_{Y}originates for Y ↔ p, the pressure tuning of a system with commentary as follows.

_{Y}, a thermodynamic pathway is regarded as expressing a state space period of 2L. The periodicity allows each λ-dependent function to be written as a Fourier series [17]. For instance, the pressure function of Figure 4 can be re-expressed as:

_{n}have nothing to do with time as with ω in Figure 1. The χ

_{n}rather identify the density of the pathway kinks governed by the program. The pathway of Panel A demonstrates twists and turns in both pressure and volume. Their Fourier representations accordingly necessitate multiple terms with diverse χ

_{n}. By contrast, an isobaric or isochoric pathway demonstrates constant p or V, respectively. In such cases, p(λ) or V(λ) require only a single term in their Fourier expressions at χ

_{o}= 0. Each representation is equivalent to that for a single state point.

**Figure 4.**Pathways and Spectral Entropy. Panels A and B show how a pathway admits a parametric representation. The representation can be expressed as a Fourier sum of trigonometric functions such as shown as in Panel C. The weight coefficients compose a power spectrum as in Panel D.

_{n}, b

_{n}determine the degree to which each Fourier term contributes. The modulus quantity:

_{n}. A plot of A

_{n}versus χ

_{n}realizes a power spectrum as in Panel D. At infinite resolution (infinitesimal Δp), Equations 8 and 9 converge to integrals which predicate an infinite number of spectral terms. A finite-step pathway is, of course, much closer to experimental reality.

_{Y}

_{↔p}that results from pressure tuning of a system: The calculation follows from the pathway of Figure 4 that is, in turn, described by the (arbitrarily chosen) parametric function:

_{n}have been rescaled by factor ξ so that:

_{n}. S

_{Y}

_{↔p}follows straightaway, viz.

_{n}and finite resolution. The logarithmic terms of Equation 14 are analogous to information surprisal quantities [18]. Each term is weighted by a normalized amplitude which is analogous to a probability term. The results of the weighted summation appear in the upper panel of Figure 5. The results for S

_{Y}

_{↔V}have been included for comparison. One observes that different thermodynamic quantities of the identical pathway need not express the same spectral entropy. The “bit” units are applicable in the same way as information. For this single, arbitrarily-chosen pathway—there are infinite possible—it requires approximately 5 bits to encode the amplitude terms in the p, V power spectra.

**Figure 5.**Power Spectra and Weighted Surprisal Sums. The lower panel illustrates the normalized power spectrum based on p(λ) of Figure 4. The upper panel illustrates the weighted sum of surprisals for pressure and volume pathway variables.

## 3. Applications and Discussion

_{Y}contributes insights in four respects. The first concerns the properties that distinguish ideal from non-ideal gases. As is well known, the former demonstrates signature features beginning with Equation 1. Additional ones include that the internal energy U and enthalpy H depend solely on N and T [6,7]. For a monatomic ideal gas:

_{V}, C

_{p}are independent of volume and temperature:

_{p}and isothermal compressibility κ

_{T}(Equation 3) are equally simple:

_{T}= 1/p. A non-ideal system requires more complicated mathematics for the state relations. The van der Waals equation is a well-established, elementary model for interacting gases [6,7,16]:

_{V}for a van der Waals system is equivalent to that appearing in Equation 17. C

_{p}does not demonstrate the same economy, however:

_{Y}

_{↔T}= 0 for isothermal pathways. Yet only an ideal gas expresses zero S

_{Y}

_{↔U}and S

_{Y}

_{↔H}for isothermal pathways.

_{Y}

_{↔CV}and S

_{Y}

_{↔Cp}. In sharp contrast, only highly select ones demonstrate zero S

_{Y}

_{↔Cp}for non-ideal systems. This is because C

_{p}depends non-trivially on p, T, V, a, and b as in Equations 23, 24, and 25.

_{Y}

_{↔αp}and S

_{Y}

_{↔κT}, respectively. Matters are more complicated for a non-ideal system. Zero S

_{Y}

_{↔αp}and S

_{Y}

_{↔κT}can only be demonstrated by highly rarefied pathways. This is because κ

_{p}and α

_{T}depend intricately on p, T, and V, and case-specific a and b. The zero S

_{Y}

_{↔αp}and S

_{Y}

_{↔κT}pathways programmed for an argon sample would not apply to neon.

_{V}or C

_{p}along a pathway that threads a range of p and T is required [6]. Usually C

_{p}is experimentally more accessible via the specific heat c

_{p}:

_{p}obtains at the price of greater pathway complexity for a non-ideal system. One way of quantifying the complexity is via S

_{Y}

_{↔Cp}.

_{p}and κ

_{T}is required at all points in a region of state space in order to realize the thermodynamic quantities U, G, S, etc. [6]. From Feature (3), one learns that it is impossible to measure α

_{p}or κ

_{T}for one state of a non-ideal system and thereby automatically know the values for points along the intersecting isotherms and isobars in the state space. As with C

_{p}, the complexity of α

_{p}and κ

_{T}is non-trivial for real systems, yet it is directly quantified by S

_{Y}

_{↔αp}and S

_{Y}

_{↔κT}.

_{rec}= 0) and adiabatic (Q

_{rec}= 0 ) transformations [1,5,6]. The first law contact with pathway spectral entropy is notably different. Specifically S

_{Y}

_{↔Wrec}and S

_{Y}

_{↔Qrec}bracket S

_{Y}

_{↔ΔU}:

_{Y}

_{↔ΔU}. There is a single exception, namely when W

_{rec}and Q

_{rec}exactly cancel at all points of a pathway; this renders S

_{Y}

_{↔ΔU}zero. More importantly, for isochoric and adiabatic pathways, (28) and (29) become equality statements:

_{Y↔ΔU }could exceed both S

_{Y↔Wrec}and S

_{Y↔Qrec}only if the exchanges were independent, thus admitting different sets of Fourier coefficients. The first law and the nature of reversible pathways preclude this. For one of infinite possible examples, Figure 6 illustrates the weighted surprisal summations that yield S

_{Y}

_{↔Wrec}, S

_{Y}

_{↔ΔU}, and S

_{Y}

_{↔Qrec}for the Figure 4 pathway. In this case, the condition in (28) holds. Evidently the complexity of programmed heat exchanges exceeds that of the work exchanges. Such a trait is not apparent from casual inspection of the pathway structure.

**Figure 6.**Weighted Surprisal Sums based on ΔU, W

_{rec}, and Q

_{rec}Power Spectra. The data derive from the pathway illustrated in Figure 4.

_{Y}

_{↔S}= 0 path is non-existent. Figure 7 shows four (of infinite possible) arrows that pinpoint nearby states. For these, there exists no S

_{Y}

_{↔S}= 0 pathway that links the initial state without expression of positive S

_{Y}

_{↔S}. The Caratheodory principle emphasizes that adiabatic pathways are exceptional for systems, ideal and otherwise. The same principle establishes the rarity of S

_{Y}

_{↔S}= 0 pathways.

**Figure 7.**Pathways and Neighboring States. The arrows point to several (of infinite possible) neighboring states that cannot be accessed by an adiabatic/isentropic path. The pathway identifies one (of infinite possible) that can link initial and final states. It is virtually always the case that changing paths incurs changes in the spectral entropy.

_{Y}

_{↔S}, S

_{Y}

_{↔V}, S

_{Y}

_{↔p}, etc. There exist neighboring pathways for which a non-zero change in S

_{Y}is impossible: Y ↔ p, T, U, μ, etc. The reason is that altering a pathway inexorably modifies one or more weight coefficients in the Fourier representation. Neighboring states that admit fixed-entropy pathways are special by the Caratheodory principle. Neighboring pathways that pose zero change in the spectral entropy prove no less special.

**Figure 8.**Carnot Cycles and Pathways. A Carnot cycle for 1.00 mole of monatomic ideal gas is illustrated in both the pV and TS planes. The Carnot strategy in relation to pathway spectral entropy is discussed in the text.

_{injected}–Q

_{wasted}[6,7].

**Figure 9.**Cyclic Pathways and Spectral Entropy. Upper and lower panels illustrate highly similar cyclic pathways for 1.00 mole of monatomic ideal gas. The volume domains are identical while the pressure domains are nearly so. The heat → work conversion efficiency is greater—25% versus 12%—for the cycle in the lower panel because S

_{Y↔TS}for the

**B**segment is less than that for

**A**.

_{Y}

_{↔T}and S

_{Y}

_{↔S}are respectively zero for isothermal and adiabatic pathways. Thus Equation 32 reflects that the optimum algorithm for heat → work conversions is where S

_{Y}

_{↔TS}is limited either by S

_{Y}

_{↔T}or S

_{Y}

_{↔S}. In effect, S

_{Y}

_{↔T}and S

_{Y}

_{↔S}establish an upper bound for S

_{Y}

_{↔TS}so as to minimize the spectral entropy. In other words, the pathway S

_{Y}

_{↔TS}must demonstrate a single source of thermal programming complexity for the maximum efficiency. Clearly all the pathway segments of Figure 8 demonstrate this critical property. The condition is unmodified if the segments are subdivided arbitrarily. Note the Carnot segments to be radically different from the pathways illustrated of the previous figures where:

_{Y}

_{↔TS}for the A and B segments immediately identifies the more efficient program. S

_{Y}

_{↔TS}is ca. 10% less for B in the lower panel; the heat → work conversion efficiency is about double that for the upper panel cycle. This holds in spite of the nearly-double temperature domain covered in the upper cycle.

## 4. Summary and Closing Comments

_{Y}distinguishes ideal from non-ideal gases; it connects as well with the first and second laws and the optimal programs encoded for heat engines. This paper focused on the pathway spectral entropy for elementary macroscopic systems. Clearly algorithms and programs are distinguished by their entropic character. Therefore a follow-up task is to identify the minimum S

_{Y}—pathways that direct a thermodynamic state toward another. Such research is currently in progress. It is further noteworthy that microscopic systems such as enzymes present thermodynamic fluctuations and pathways under dynamic, non-equilibrium conditions [20,21]. The pathways of these systems might benefit from an analysis in spectral entropy terms. The properties described here for non-ideal systems, and first and second law constraints offer valid starting points.

## Acknowledgements

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**MDPI and ACS Style**

Graham, D.J.
On the Spectral Entropy of Thermodynamic Paths for Elementary Systems. *Entropy* **2009**, *11*, 1025-1041.
https://doi.org/10.3390/e11041025

**AMA Style**

Graham DJ.
On the Spectral Entropy of Thermodynamic Paths for Elementary Systems. *Entropy*. 2009; 11(4):1025-1041.
https://doi.org/10.3390/e11041025

**Chicago/Turabian Style**

Graham, Daniel J.
2009. "On the Spectral Entropy of Thermodynamic Paths for Elementary Systems" *Entropy* 11, no. 4: 1025-1041.
https://doi.org/10.3390/e11041025