C.1 Spontaneous Negative Entropy: Growth and Repair

While devices and systems that we use every day will not spontaneously repair themselves, Mother Nature has apparently provided life forms with this capability. In fact, we can think of growth as a sort of negative entropy which occurs in the first part of our lives. The overall growth or repair process must still generate positive entropy. Still, in our definition of entropy for a system, we understand spontaneous positive entropy change ΔS > 0 as the amount of disorganization that occurs; then spontaneous negative entropy change

(C.1)

is a term that we might argue can apply to Mother Nature’s life forms. In human life, as we grow we become more organized; we have a larger capability for doing more useful work so our free energy is essentially increasing. In a sense we can label this as spontaneous negative entropy. Furthermore, when we are injured, our bodies will try and repair the damage by creating a spontaneous amount of negative entropy equal to or greater than the entropy damage (see Equation (1.13)) change that occurred during our injury. This quantity, in thermodynamic terms, is estimated in Equation (1.14).

Negative entropy was first introduced by Erwin Schrödinger in a non-technical field in his 1944 popular-science book What is Life [1]. Schrödinger uses it to identify the propensity of the living system to want to organize, which is contrary to the second law. That is, for most of us, we like to build houses, build cities, and organize our way of life. This is also observed in lower life forms. So this book’s discussion on the subject was limited to such areas. The subject discussed here is not found easily in the literature and is an atypical thermodynamic term as we have described it.

It is not immediately obvious why spontaneous negative entropy processes would be allowed by the second law of thermodynamics.

The spontaneous tendency of a system to go toward thermodynamic equilibrium cannot be reversed without at the same time changing some organized energy, work, into disorganized energy, heat.

Some would argue that repair and growth are not spontaneous negative entropy. Certainly repair does not reverse time; repair is done by removing the damaged area and re-grows the cells as close to their original growth state as nature permits. However, growth and repair are not easily controlled either which makes them more or less spontaneous events. So, from this perspective at least, we can think of it as spontaneous negative entropy. The key phrasing of the second law states that there is a natural tendency “to come to thermodynamic equilibrium.” Mother Nature likely creates a closed system that encourages growth and repair in order for the system to come to some sort of final growth or repair equilibrium condition. This means that creating a closed system takes energy, that is, part of the disorganized entropy production. In the non-equilibrium state we can think of the organized growth or repair as

(C.2)

Growth or repair then stops when the system is in thermodynamic equilibrium or perhaps a quasi-equilibrium state. Growth and repair must obey the second law. Therefore, the overall entropy production is positive, yet the results are organized matter.

But what is being disorganized then that occurred in the repair or growth process? The growth and repair processes are similar to that discussed in Section 1.9.1. Apparently, it can be thought of like the repair of a device: the system undergoes a seemingly negative entropy change, but with processes that must have a tendency to come to thermodynamic equilibrium. Therefore entropy is being maximized in growth but with the results of organized matter; overall, the universe has a positive entropy change equal to or greater than the negative entropy change. Energy has been used to create and repair the living system. Mother Nature has converted fuel (food) which becomes decomposed to build or repair our bodies. The process is similar to repair of a device, in that the body is manufacturing cells for growth and repair. The unused decomposed mater is waste that adds entropy to the universe. A propose model is provided here.

C.2 The Perfect Human Engine: How to Live Longer

In this section we would like to make use of some of our knowledge that we have built up in thermodynamic degradation and see if we can apply it to the miracle of the human engine. Although this is not the perfect science for such a task, as Mother Nature is very complex, we can use the opportunity to exemplify some strong analogies that exist as well as some major differences. To this end, we can gain insight that can help us manage our human engine and perhaps live a little longer.

If we were to invent the perfect human engine, what characteristics would it have? Our engine objectives might be: high work efficiency; long life; and the ability to repair damage close to near-perfect original conditions.

C.2.2 Knowledge of Cyclic Work to Improve Our Chances of a Longer Life

Some of the thermodynamic analogies at this point may seem obvious to improve human engine life, but they are worth noting. Clearly, irreversible damage reduces efficiency. As noted in Equation (2.88):

(C.3)

Damage is an enemy as it creates irreversibility and reduces our ability to change fuel into work. Although some damage is reversible due to repair, repair inefficiencies also occur and irreversibilities build up over time creating aging. Inefficient fuels are also poor choices, creating inability to convert fuel into work and causing excess waste.

In terms of cumulative damage, we have noted that stress is a key factor in increasing work cycles to failure. Some examples of stress for the human engine cycles are as follows.

• Digestive cycle: Environment food-stress might include hard-to-digest foods (processed foods); large-sized food not initially reduced in size (suggesting pureed food may be helpful); high-stress cycles such as eating large quantities of food at one time (excessive overeating).
• Body work cycle: high-stress cycles include activity that can cause damage (football, excessive exercise, extreme sports); and cyclic interruption, abrupt change to daily work, daily rest and repeat.
• Heart cycle: high-stress cycle for long periods such as marathon training compared to light joggers [2].
• Breathing cycle: air quality can cause stress; excessive athletic use cycles (e.g., marathon running).
• Brain cycle: change in daily cycle of mind work, resting and repeating; excessive work load; high emotional stress.
• Repair cycle: damage–repair daily regeneration cycle; excess damage loads and lack of nutrients, oxygen, and blood flow can increase repair cyclic stress.

These ideas are summarized in Table C.1. Some ideas may be counterintuitive such as high-stress exercising. One study on the human heart suggested a U-shaped stress versus life cycle curve that we discuss in the following section. What we know from cyclic work and Miner’s rule can seem at times related to human life cycle; reducing stress loads can increase longevity; inactivity also reduces life span, however. Thus, a U-shaped S–N curve seems appropriate. In other areas there may be more parallels to engines. For example, there have been suggestions that eating less but more frequently would be beneficial. In cyclic theory this makes sense. Table C.1 provides a cyclic summary.

Human engine cyclic subsystem	Local interacting environment	Work cycle	High-amplitude stress cycles
Digestive system	Food	24-hour daily cycle of chewing, energy conversion digestion, waste	Hard to digest foods, large size foods, overeating
Body work cycle	External work	24-hour daily workloads: rest, repeat	High stress activity interruption, abrupt change to daily work
Heart cycle	Blood, air	50–100 beats/min, expansion–contraction work to circulate blood	High stress activity and lack of oxygen contribute to a shorter life; moderate exercise is better than heavy exercise
Breathing: lungs cycle	Air	About 12 breaths/min, inhale–exhale	Air quality excessive or unusual exhaustion
Brain cycle	Five senses	24-hour daily cycle, awake–sleep–other	Change in daily cycle of mind activity, high stress on one or more of the five senses, excessive work load, high emotional stress
Repair cycle	Body, blood, oxygen, nutrients	Damage-repair, daily regeneration	Poor blood flow and lack of oxygen and nutrients creates additional stress, excess damage due to trauma injury

C.2.3 Example C.1: Exercise and the Human Heart Life Cycle

The reader may be skeptical of the results related to high stress activity on the heart, as exercise and lifespan are correlated. However, the following is the result of a Copenhagen City Heart Study by Schnohr et al. [2]. A total of 1098 healthy joggers and 3950 healthy non-joggers were followed from 2001 with the study published in February 2015.

“[Results:] Compared with sedentary non joggers, 1 to 2.4 h of jogging per week was associated with the lowest mortality (multivariable hazard ratio [HR]: 0.29; 95% confidence interval [CI]: 0.11–0.80). The optimal frequency of jogging was 2 to 3 times per week (HR: 0.32; 95% CI: 0.15–0.69) or less than or equal to 1 time per week (HR: 0.29; 95% CI: 0.12–0.72). The optimal pace was slow (HR: 0.51; 95% CI: 0.24–1.10) or average (HR: 0.38; 95% CI: 0.22–0.66). The joggers were divided into light, moderate, and strenuous joggers. The lowest HR for mortality was found in light joggers (HR: 0.22; 95% CI: 0.10–0.47), followed by moderate joggers (HR: 0.66; 95% CI: 0.32–1.38) and strenuous joggers (HR: 1.97; 95% CI: 0.48–8.14).

“[Conclusions:] The findings suggest a U-shaped association between all-cause mortality and dose of jogging as calibrated by pace, quantity, and frequency of jogging. Light and moderate joggers have lower mortality than sedentary non joggers, whereas strenuous joggers have a mortality rate not statistically different from that of the sedentary group.”

At least in this one study, verification seems to indicate that Miner’s type rule for human mortality very roughly has some validity. Obviously, it is clear that low stress will not lead to longer life as in the case of metal fatigue. The study suggested a U-shaped model. We have taken their U-shaped description as an initial idealized S–N curve model here (not given in their study) of the type plotted in Figure C.1 where light to moderate stress increases life cycle expectancy for the human heart. The U-shaped curve is just meant to illustrate the major difference between human fatigue and fatigue in metals. It is reasonable to assume that a major difference between metal fatigue damage versus human damage is the repair factor, which is discussed in the next section. If we can completely repair our daily damage, then we could live forever. On the other hand, if we could not repair ourselves, then it is likely that human heart fatigue would follow closely to the metal fatigue line. We know that the ability to do repair is also related to exercise; without it, the body cannot properly do its repair job. Yet too much exercise will cause excess damage that the repair cycle cannot keep up with (Figure C.1).

C.3 Growth and Self-Repair Part of the Human Engine

Growth and self-repair indicate significant differences between the human engine and a normal engine or device. First of all, growth is unlike what is expected from the second law of thermodynamics. We can say that the human engine in the early part of life is becoming more ordered, and its free energy is actually increasing in the sense that the amount of useful work that the human engine can perform increases as we grow older. We then have a case that for growth the human engine is becoming more ordered during the growth phase, and entropy must be decreasing:

(C.4)

Furthermore, when damage occurs, disorder is created in the human system; repair then reverses damage with a certain amount of efficiency. However, prior to repair, damage increased entropy:

(C.5)

Then when repair starts, damage becomes somewhat reversible:

(C.6)

The exchange of entropies in repair is

(C.7)

The total entropy of any process increases; in keeping with the second law (Equation (1.4)) . Since , than the entropy damage to the environment must be positive and greater than the negative repair entropy by the second law, that is:

(C.8)

(Also see Section 1.9.1.) Since the repair process is in itself cyclic, this means that the internal energy needs to be restored to its original state. Its change due to the damage portion of the cycle is:

(C.9)

We can make a repair model. In a simplified view, the body is essentially a battery that gets charged in the repair process by the amount above. Since it must discharge nutrients adding heat and work to repair, the internal energy change is essentially in a damage–repair cycle by the combined first and second law (see Equation (2.60)):

(C.10)

That is, for a perfect repair the internal energy in the cycle is unchanged; an imperfect repair leaves us with some inefficiency and permanent change to the internal energy, however. Daily use causes damage, which also requires repair. The unrepaired portion builds up, causing body fatigue and damage.

C.3.1 Example C.2: Work for Human Repair

We can make a simplified thermodynamic repair model for instructional purposes. The repair process is shown in Figure C.2. An injury occurs, and after a few hours the entropy is at a maximum where the entropy damage is S_damage and the area is infected with temperature rise T_high. Repair work is done W_repair and the injury is almost completely repaired; the unrepaired entropy is S_unrepaired. The change to the internal energy from repair cycle ΔU is due to the unrepaired damage. In this case from the first and second law the minimum repair work is:

(C.11)

The negative entropy generated for the repair process is

(C.12)

where S_repair is the negative entropy needed for the repair process. That is, in the case of repair:

(C.13)

so that the minimum work in the repair process is found by combining these equations

(C.14)

In the case of perfect repair S_damage = −S_repair and the minimum repair work is

(C.15)

where Q_heat is the heat dissipated prior to repair which can be measured.

The efficiency of repair is

(C.16)

In the case of perfect repair where , the efficiency is 1. Since , a fraction f of S_damage, then their ratio (S_repair/S_damage) will give a value between 0 and 1, and this obeys the relation in Equation (3.15) where

similar to a cyclic heat engine process.

C.4 Act of Spontaneous Negative Entropy

When we repair or manufacture a device, the device becomes more organized so the entropy of the device has decreased. But the process to build or repair caused excess entropy to the universe. In the same way, energy is used to create and repair a living system. Mother Nature uses some fuel (food) for energy needed for growth and/or repair, and the food decomposes into waste which is an increase in entropy production. The decomposed matter is waste that adds entropy to the universe. This helps to justify spontaneous negative entropy as not violating of the second law. The act of spontaneous negative entropy can be clarified somewhat as follows.

The act of negative spontaneous entropy found in living systems is the result of initial work done by the living system to create a tendency for the living system to come to equilibrium with its neighboring living environment that drives the living system to a state of repair or growth in order to achieve thermodynamic equilibrium. The overall process of entropy production generated between the environment and the system is still positive, in keeping with the second law.

If we concur with this statement, where we are trying to maintain agreement with the second law to have a tendency to come to thermodynamic equilibrium, it means that our human engine must create an environment that encourages spontaneous negative entropy. How does this come about? We can hypothesize that our body takes in food and converts it to work, charging up an area that drives the need for repair or growth. In this sense there is not much difference between repair and growth. We know that an injured area will not reverse itself and become organized, seemingly going back in time; it must be removed and re-grown. This is similar to device repair.

The key phrasing of the second law is then: there is a natural tendency “to come to thermodynamic equilibrium.” Then we can surmise that Mother Nature creates a closed system that encourages growth and repair in order for the system to come to some sort of final equilibrium condition of full growth or full repair. In the non-equilibrium state:

Growth or repair then stops when the system is in thermodynamic equilibrium or perhaps a quasi-equilibrium state.

In order to improve our understanding and remain within the scope of this book, we can hypothesize part of the equilibrium process from our prior models in Chapters 1 and 8. Once the nutrients are brought to the repair site, they likely diffuse into the areas under diffusion forces. Diffusion can be driven both by a concentration gradient and an electrical charge across the repair area [3]. This is one possible scenario, that is, we can hypothesize something like the equation below where we have combined Chapter 2 (Equation (2.75)) and Chapter 8 (Equation (8.8)) results to obtain

(C.17)

Here energy flow will go from the higher temperature area so that the repair energy is increasing . If then and the repair area is losing charge; for , the repair area is receiving particles. When , , and , the repair process is completed and we are in thermodynamic equilibrium. The result is a more organized area. The environment in a sense appeared to increase entropy, yet the repair or growth area became more organized. In this model, the body is tricked into increasing entropy change dS_total, that caused more organization dS_repair than disorganization in order to come to equilibrium with the neighboring environment.

This is just meant to demonstrate how a spontaneous repair or growth could be taking place in a living system in agreement with principles of the second law. There are so many unanswered questions, such as how does the body know what cells match for repair and how does the body set up a spontaneous tendency for repair and growth?

C.4.1 Repair Aging Rate: An RC Electrical Model

We might assert that the rate of negative repair entropy varies as the entropy damage to within an aging factor f(t):

(C.18)

For increasing aging, we propose that f(t) increases where 0 < f(t) < 1. Therefore, the rate of change of f(t) is some function of the unrepaired entropy damage S_unrepaired(t) that builds up over our lifetime and reduces our ability to heal at an older age. As we grow older, our ability to heal completely is reduced and f(t) changes. If this was not the case and we had complete repair every day, we would not age. As f(t) decreases, then according to this proposed equation the repair rate will also slow. The model is simple but illustrates a possible view of aging. From our experience, f(t) must be a slow function of time compared to the repair rate, that is,

(C.19)

We can therefore write

(C.20)

and treat f(t) as a constant over the repair time period when we look at the entropy repair rate.

Equation (C.18) can be compared to the well-known RC circuit shown in Figure C.3. The notion that the body charges up (switch B) in order to energize the repair area, before discharging energy (switch A), has a similar differential equation given by

(C.21)

Comparing this model to Equation (C.18), S_repair = –S_damage and charge flow equates to entropy flow. The fractional repair and repair time are f(t) ~ 1/R(t)C(t) and I is then the entropy current. For f(t) to decrease, R and C would need to increase. We would need at this point a biological model for R and C. Perhaps it makes sense that the resistance of the internal body increases with time as the unrepaired entropy disorder builds up over a person’s lifecycle. After all, we associate entropy production with resistance. C would also increase if the dielectric constant increases; this needs some thought on why that would occur if disorder were to increase over time. It may be that only R increases and C is constant.

References

1. [1] Schrödinger, E. (1944) What Is Life? Cambridge University Press, Cambridge.
2. [2] Schnohr, P., O’Keefe, J.H., Marott, J.L., Lange, P. and Jensen, G.B. (2015) Dose of jogging and long-term mortality. Journal of the American College of Cardiology, 65 (5), 411–419.
3. [3] Becker, R.O. and Selden, G. (1985) The Body Electric, Electromagnetism and the Foundation of Life, William Morrow Publisher, New York.