Irreversibility As Thermodynamic Time

Open Physics 2021; 19: 127–133

Research Article

Charles C. Hwang* Irreversibility as thermodynamic time

https://doi.org/10.1515/phys-2021-0018 Keywords: time, availability, irreversibility, time dilation, received August 30, 2020; accepted February 21, 2021 gravity, entropy Abstract: In Newtonian mechanics, time and space are perceived as absolute entities. In Einstein’s relativity theory, time is frame dependent. Time is also affected by gravitational field and as the field varies in space, 1 Introduction time also varies throughout space. In the present article, the thermodynamic-based time is investigated. In macro- 1.1 Time in classical mechanics scopic view of thermodynamics, energy is conserved in every system or process. On the other hand, exergy fi (availability) is not conserved and can be destroyed, In Newtonian mechanics, the time in the universe is xed [ – ] and “irreversibility” is generated. Since each thermody- regardless of location or epoch 1 4 . This means that all fi namic system may generate different amounts of irrever- events can be regarded as having a distinct and de nite sibility, this quantity is system dependent. The present position in space and occur at a particular moment of article investigates the characteristics of entity irreversi- time. Time as one perceives is absolute and seems to fl bility. (1) It is found that the entity behaves in the similar ow steadily and uniformly regardless of anything external. manner as the clock time in the standard configuration of This moment of time is taken to be the same for observers - inertial frames under Lorentz transformation. (2) It is also everywhere in the universe. Time is an un stretchable found that the entity is affected by gravity fields in the quantity, in terms of which, changes in the whole universe - similar manner as the clock time. We have demonstrated could be uniquely described. The theory constructs a deter that, like clock time, irreversibility is frame dependent, ministic set of mathematical relations that allow prediction and affected by gravity in the similar manner as the clock of the future and past behaviors of moving objects. All that time. For these reasons, we propose to call the irreversi- one needs in order to do this are data in the present bility of the system as the thermodynamic time. The time’s regarding these moving objects. Equations of motion in arrow is automatically satisfied, since irreversibility gene- Newtonian mechanics are invariant under T. ration always proceeds in one direction (toward future). Based on the strength of the findings (1) and (2),apossible application of the irreversibility is an interpretation and management of the aging of biological systems. It is shown 1.2 Time in relativity theory by other authors that entropy generation (equivalent to irreversibility) is a parameter for the human life span. In the special and general theories of relativity, time is ’ Our sensation of time flow may be attributed to the flow stretchable and varied from place to place. Einstein s - of availability and destruction of it through the living mechanics indicates that the time passage for two indi system. viduals moving relative to one another, or experiencing a gravitational field, is different [1,2,5–7]. These theories finally break Newtonian mechanics’ rigid conception, though the flexibility of time passage becomes apparent only at high speeds or in strong gravitational fields.  In the following subsections, the rates of clocks differ * Corresponding author: Charles C. Hwang, Department of (1) on moving frames with a constant relative velocity and Mechanical Engineering, University of Pittsburgh, Pittsburgh, ( ) fi ff PA15261, USA, e-mail: [email protected] 2 on uniform gravity elds with di erent strength. In Professor Emeritus; 3005 S. Leisure World Blvd. # 404, Silver later sections, it will be shown that a thermodynamic Spring, MD 20906-8305 quantity irreversibility, I, behaves like time.

1.2.1 Standard configuration [8] their frame. The difference in the rates of two clocks is calculated next. The following frame arrangement is defined for use in ΔΔ1tt′ β22 Δ1 t 1 βt Δ, future discussions. =−=−(−−) Imagine that two rigid reference frames F and F′ are 1 v2 ΔΔtt− ′ =( 1 − 1 − βt2 ) Δ ≃ Δ,t (1.6) in uniform relative motion with velocity v. For both 2 c2 frames, identical units of length and time are used. neglecting magnitude of fourth and higher order [6].If Their time tt, ′ and their Cartesian coordinates x,,y z v =/10 m s , 1/(25.56310vc22 / )= × − 16, showing that the and ′′,,′ form the coordinate systems F :,,,xyzt x yz { } time dilation is very small with low velocity v. and F′ :,,,{xyzt′′′′}. The systems are said to be in standard configuration, if they are arranged in the following - way. The origin of F′ frame moves with velocity v along 1.2.3 Rates of clocks in gravitational field the x-axis of F, the x′-axis coincides with x-axis, while the y- and y′-axes remain parallel, so do the z- and In a uniform gravity field, a clock h sitting on a high shelf z′-axes; and all clocks are set to zero when the two ori- will run faster than a clock ℓ on the floor. If clock h is the gins meet. emitter of light with frequency ωh and clock ℓ is the receiver, Feynman [5]¹ finds the frequency at the receiver

as follows. A photon of frequency ωh has energy εωh =ℏ h. 2 1.2.2 Rates of clocks in standard conﬁguration Since emitted energy εh has the gravitational mass εch/ 2 the photon has a mass ℏωch/ and is attracted by the

In special relativity, the Lorentz transformation equa- earth. In falling the distance HH=−h Hℓ it gains an addi- 2 tions for the primed and unprimed variables in standard tional energy (ℏ/)ωch gH, so it arrives at the receiver with conﬁguration are as follows: the energy

xvt−  gH  x′ = , (1.1) εωωℓℓ=ℏ =ℏh 1 + . 1 − β2  c2 

y′ = yzz,,′ = (1.2) We rewrite the equation as

tvxc−/2 ()Rate at the receiver ωℓ t′ = , (1.3) ( ) 1 β2  gH  1.7 − =(Rate of emissionωh ) 1 +  .  c2  where β =/v c. The inverse of the last equation can be If we write the equation in terms of the time rate written as tvxc′ + ′/ 2 1 1  gH  t . ( ) 1  = 1.4 =+2 1 − β2 Δttℓ Δ h  c 

Einstein has calculated the time dilation under the or standard configuration. This proves that time is frame- Δt gH h 1 . ( ) [ ] =+ 2 1.8 dependent 6 . Consider two inertial frames F and in Δtℓ c standard configuration. The frame F′ moves at velocity For H 20 m, gH/= c2152.182 × 10− , that is, for an v relative to the frame F along the x-axis. Consider a = altitude difference of 20 m at the earth’s surface, the clock at rest in F′. Let two events be, 1 and 2, which occur time difference, ΔΔtt− , is only about two parts in at the same point x′ = x′ in F′, indicated by the interval as h ℓ 21 1015. This result proves that time depends on the strength Δttt′ = ′ − ′. Substituting these values in equation (1.4) 21 of a gravity field. Note that the quantity gH/ c2 is non- yields dimensional. ΔΔ1tt′ =− β2 , (1.5) where Δtt=−2 t1. Observers in F observe that the moving clock is running slow. This effect, time dilation, is reci-  procal. That is, if a clock is at rest in F, observers in F′ 1 Feynman derived the result with the help of the principle of find that it runs slow as compared with clocks at rest in equivalence. g is assumed constant in this thought experiment. Irreversibility as thermodynamic time  129

2 2 Availability and irreversibility  T  Δ10 δQ W p Δ, V AI=−∫   −(−)−0 ( )  Tb  2.5 The conservation of energy for a control mass can be 1 expressed as [9,10] heat transfer work interaction irreversibility

dEδQδW=−. (2.1) where ΔΔA =+EpVTS00 Δ − Δ. The term irreversibility I(=Tσ) accounts for the destruction of availability due The entropy equation can be written as 0 to irreversible processes within the control mass. δQ Equations for flow availability can be written for an d,S =+δσ (2.2) T open system, including heat transfer and matter flow at [ ] where σ is the entropy production which can be stated as: the boundary surface 9,14 . δσ > 0 internally irreversible process, δσ = 0 internally reversible process, δσ < 0 impossible process. 3 Special features of irreversibility Aquantityavailability (also called exergy, work potential, Arbeitsfähigkeit) has been used by engineers to assess the maximum work that can be obtained for a 3.1 Irreversibility and frame dependence combined system, closed or open, in a given environ- mental condition [9,11,12]. It is known that all macro- We shall demonstrate that the irreversibility I is frame processes are irreversible, and the availability is destroyed. dependent, similar to clock time. Planck² and Einstein³ The destruction of availability is called irreversibility. considered thermodynamic systems which are in the Wall [13] stated that energy is motion or ability to standard configuration [10,15]. First, we note that irrever- produce motion and exergy is work or ability to produce sibility can be expressed as I = T σ [9]. In the standard work. He also stated that time is experienced when 0 configuration of frame F and frame F′, the quantities exergy is destroyed, i.e., an irreversible process, which T and σ are related by the following equations when creates a motion in a specific direction, i.e., the direction 0 ( of time. The idea that availability can be destroyed is Lorentz transformations are applied separately p. 159, )[ ] useful [9]. This concept essentially brings together the Tolman 10 .

ﬁrst and second laws. T0F′ = T0F, (3.1) If the state of a thermodynamic system departs from 1 − β2 that of the environment, an opportunity exists for devel- σ = σ , (3.2) oping work. For a combined system of a control mass FF′

(closed system) plus the environment, the work Wc for which implies that σ is invariant under Lorentz transfor- the combined system is given by [9] mation in the standard conﬁguration. Then

WEUpVVTSSTσ=( − )+ ( − )− ( − )− . (2.3) 2 cc00 00 00 I ′′′′Tσ Tσ v F ==0FF 0FF 11 −=−β2 (3.3) Tσ Tσ c2 Since Tσ0 c is positive when irreversibilities are present IF 0FF 0FF and vanishes in the limiting case where there are no v2 irreversibilities, the maximum theoretical value for W is IFFFF′′=−1impliesIII <. c c2 obtained by setting Tσ0 c to zero in equation (2.3): The diﬀerence in value is ( ) A =WEUpVVTSSc,max =( − 0 )+ 0 ( − 0 )− 0 ( − 0 ). 2.4 v2 IIFF−=−−′′ I F1 I F The function A is called availability for the control mass. c2 To sum up, the availability is deﬁned as the maximum  v2  theoretical useful work obtained if a system is brought 11 (3.4) =−−IF  2  into thermodynamic equilibrium with the environment  c  by means of processes in which the system interacts 1 v2 ≃ IF, with this environment [14]. In the general case, a control 2 c2 mass may experience both heat and work interactions - with other systems, not necessarily including the envir  onment, and there is some availability destruction within 2 Plank, Berl. Ber. 1907, p. 542; Ann. der Physik, 26,1(1908). it during such interactions. 3 Einstein, Jahrb. der Radioaktivität und Elektronik, 4, 411 (1907). 130  Charles C. Hwang

2 neglecting magnitude of fourth and higher order. Compar- where we make approximation EEEmcℓℓ−≈≈0 , and ing this result with equation (1.6), we see that “time” and finally obtain irreversibility behave similarly, i.e., I gH h ≈+1 . (3.9) 2 2 Δt′ 1 v I ′ Iℓ c =−1 = F . (3.5) Δt 2 c2 IF Comparing equation (3.9) with equation (1.8), we observe the similarity between time and irreversibility in the pre- sence of gravity field, i.e., Δt gH I 3.2 Irreversibility in a gravitational field hh1 . ( ) =+2 ≈ 3.10 Δtℓℓc I In this section, we analyze the effects of a gravitational field on the value of irreversibility I. The objective is to compare the result of this thought experiment with the gravitational field on “time” as described in Section 1.2.3. 3.3 Clock time versus thermodynamic time We consider the irreversibility of a closed system with mass m as it is lowered from height zH= h to height After collecting the results from Sections 3.1 and 3.2, we z = 0 (datum). Equation (2.5) will be used. This process shall examine the characteristics and implication of the has no heat transfer δQ, no work W, no change in the entities irreversibility I and clock time Δt. Now we have volume ΔV, and no entropy change ΔS, so that two entities Δt and I, both of which behave similarly under Lorenz transformation in one case (equation (3.5)) ΔorΔAI=− I =− A. and under gravity influence in separate case (equation This implies that during the process, availability is (3.10)). It is proposed that Δt be called “clock time” and destroyed. The kinetic energy is not considered, so that I be called “thermodynamic time.” It appears that Δt is E consists of internal energy U and potential energy mgz, more abstract and theoretical and that I is more tangible fi assuming uniform eld. Denoting Ih as the irreversibility and operational [16–18]. The research on clock time is generation for lowering the system from zH= h to z = 0, being continued: Barbour has described timeless quantum ( ) the above equation gives g is assumed constant cosmology, and Smolin has argued evolutionary time.

Ihh=−ΔE Currently, Δt can be measured with utmost precision and EE I can be measured with as high degree of precision as =−(0 −h ) ( ) 3.6 fi =−[(UUmgH0 +0 )−(hh + )] energy measurement. It can be expected that I will nd fl = mgHh, application in which energy ow is involved. 1. In describing time, what we are conscious of is a ( ) assuming Uh = U0 mainly a function of T . Next, we memory of the past tinged with an expectation of the consider the irreversibility of the same system which is future [19]. Once the occurrence of an event has lowered from height zH= ℓ to the height z = 0, where passed, the event has become a memory and cannot HHℓ < h. Denoting Iℓ as the irreversibility generation for be repeated or corrected. In the words of Bohm [20]: lowering the system from zH= ℓ to z = 0, we have “Although the present is, it cannot be specified in ” - Iℓℓ=−ΔE words or thought without slipping into the past. Pen =−(EE − ) rose [21] stated that central to our feeling of awareness 0 ℓ (3.7) =−[(UUmgH0 +0 )−(ℓℓ + )] is the sensation of the progression of time. Based on ( - = mgHℓ, these dialogs which come from human biological sys tems), time appears to be the energy (in this case, assuming again Uℓ = U0. The irreversibility generation of irreversibility generation) flowing through the biolo- lowering the system from Hh to Hℓ is gical system. The future is an untrailed territory that

Ihh−=IℓℓℓmgH − mgH =(−)= mg H h H mgH. is hidden behind a mist. 2. The entity I is always positive and zero in the limit. Next, dividing both sides by I yields ℓ The notion of irreversibility presupposes “time’sarrow,” II− I mgH mgH gH hhℓ 1,( ) because the direction of the process is always in the =−== ≈ 2 3.8 IℓℓII ℓℓEE− 0 c direction toward future. Time is asymmetrical with Irreversibility as thermodynamic time  131

respect to the event axis (world line).Thisiscom- irreversibility (or entropy because I = T0σ). This signifies monly called the arrow of time. as degradation of the useful energy of the system, and we Our experience shows that time is asymmetric to have interpreted this as the passage of time for the the past and the future, and analogous to the flight of system, i.e., the system possesses its own time. an arrow. This means that if we arrange events in the For inanimate systems, irreversibility can also be chronological order along an axis, the time appears generated. This includes oxidation, erosion, or heat transfer. to be asymmetric with respect to the axis. This also Some inanimate systems, such as diamond or gold, may means that the events are irreversible; the events produce irreversibility at slow rates. This implies that the cannot be played back as a movie playing backward. change of the state (time) proceeds slowly for these systems. The physical basis of the direction of time has The irreversibility comes from heat transfer, diffu- been discussed by many authors [2,16, 18,22–27]. sion, fluid viscous dissipation, and chemical reactions, There appears to have arrow of time of different ori- which may operate at the cellular level. Some examples gins: the thermodynamic arrow of time, biological of irreversibility production are as follows: arrow of time, the arrow of time of retarded electro- 1. Heat transfer – The irreversibility production comes magnetic radiation, cosmological time asymmetry, and among bodies with finite temperature differences [9]. others. For the thermodynamic arrow of time, Smolin 2. The irreversibility production in pipe flow in which [16] stated that small bits of the universe, left to them- the temperature of fluid is different from that of the selves, tend to become more disordered in time (the pipe [28]. spilt milk, the air equilibrating, and so on).Itisnoted 3. Diffusion of fluids of different types produces irrevers- that our direct contacts with the nature follow macro- ibility. scopic thermodynamic laws, and the thermodynamic 4. Combustion of various types produces irreversibility [9]. arrow of time prevails. 5. The irreversibility production of flowing viscous fluid Good examples showing the asymmetry of time in a pipe [28]. are biological systems. All biological systems are sub- ject to continuous irreversible changes through their The system in these examples may be either biological or entire lives. There is a type of people called progerin inanimate. who grows much faster than ordinary people. This indicates that time passage depends on individual systems. 3. According to the Copenhagen interpretation of quantum 4.1 Application of thermodynamic time in mechanics, quantum evolution is governed by the human aging Schrödinger equation, which is time symmetric, and by wave function collapse, which is time irreversible. Silva and Annamalai [29,30] correlated life-span entropy Despite the post-measurement state being entirely generation and the aging of human. The organs consid- stochastic in formulation of quantum mechanics, a ered are brain, heart, kidney, liver, adipose tissue, ske- link to the thermodynamic arrow has been proposed. letal muscles, and the rest of the organs. To estimate Thus, the modern physical view of wave function col- entropy generation during a human life span, an avail- lapse, the quantum de-coherence, the quantum arrow ability analysis is applied to the metabolic oxidation of of time is a consequence of the thermodynamic arrow the three main nutrient groups: carbohydrates, fats, and of time [27]. proteins, in order to obtain the entropy generated for each of them under isothermal conditions. Entropy generated over the life span of average individuals (natural death) was found to be 11,404 kJ/K per kg of body mass. The entropy generated predicts a life span 4 Irreversibility of various systems of 73.78 and 81.61 years for the average US male and female individuals, respectively, which are values that Once the similarity of irreversibility I and clock time is closely match the average life span from statistics (74.63 discovered, we may want to find its applications. A ther- and 80.36 years). These articles assume that entropy gen- modynamic system may be closed or open, biological or eration is the mechanism for biological aging, and that inanimate. The system interacts with its environment and life comes to an end when entropy generation reaches its in the process destroys availability and generates maximum. 132  Charles C. Hwang

The articles cited here provide additional and tan- 3. Irreversibility generated, MJ/kg: male, 3,567; female, gible evidence for our discovery that the irreversibility 3,503 and clock time are intimately related. The entropy gen- 4. Equivalent human age, year: male, 74.63; female, eration σ in these articles is similar to the irreversibility 80.36 generation I(=Tσ0 ), but entropy is invariant under 5. I/m Jkg/ s: male, 1.52; female, 1.38 Lorentz transformation. As indicated by Denbigh [31], the concept of irreversibility has a much wider ﬁeld of These results may be interpreted as that a human male destroys about 1.5 Joule of useful energy per kg body application than has the concept of entropy generation. weight every second in his life span. In addition, irreversibility has the dimension of energy, which may be easier to visualize and control.

5.2 Impacts and implications of 5 Concluding remarks thermodynamic time

We have two stems of time, i.e., thermodymanic time and The contribution of the present article is the derivation of clock time, both behave similarly under Lorenz transfor- equations showing that irreversibility I behaves like mation and a gravity field. These times are local and time (1) in the standard configuration of inertia frames system dependent. The overall picture is that as the avail- under Lorentz transformation, and (2) I is affected by ability (exergy) of the system is destroyed, the thermo- gravity fields in the similar manner as the clock time. dynamic time progresses, like the progress of the clock As pointed out previously, thermodynamic time I has time. As we know, the irreversibility of all systems the dimension of energy. Many phenomena in biological increases. If the system is biological, we may call this systems where duration is involved, such as aging, bio- aging. logical clocks, and defects in biological clocks (Parkinson’s disease), may be closely related to energy flow through irreversibility. Our sensation of time flow may be explained as the availability flow and destruction of it (irreversibility), 5.1 Thermodynamic time and its application in the body and brain. Our understanding of these problems is in the infant stage, and further research is In Section 3.2, we have seen the calculation of irreversi- required. Since irreversibility is based on a thermody- bility for a closed system. If we desire not to have the size namic concept which is universal, further application to (mass) of the system as a parameter, we may use I/m,so other fields, such as cosmology (the Big Bang, infla- that the size of the system may not become a parameter tionary cosmology, and black holes), is possible. and comparison between various systems can be per- formed, i.e., I ΔτC= , m Nomenclature where τ is a thermodynamic time and C is a dimensional c velocity of light constant. The treatment of human aging by Silva and E energy [ ] Annamalai 29,30 cited in Section 4.1 has used open F, F′ coordinate systems systems. Since for an open system mass and energy are g acceleration of gravity fi not xed, analysis is more complicated. Here, we may use H height separation the instantaneous values of I/m, I/E,orI/A within the m mass control volume. p pressure It is natural to ask how I and Δt are related. From the Q heat energy [ ] work of Silva and Annamalai 29 , the following numbers q″ heat flux per unit area can be listed. S entropy 1. Entropy generated, kJ/kg K: male, 11,508; female, T temperature 11,299 t time 2. T0 = 37° C= 310 K U internal energy Irreversibility as thermodynamic time  133

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