The probabilistic concept is used when the actions of the living beings are integrated into a memory structure to later use it as an advantage, and thus increase the probability of living.

In this work, we assume that the probability of individual survival is a monotonic function of time (t), which suddenly goes to zero at the time T when the characteristic lifetime of a biological species is reached.

Furthermore, it is assumed that any biological system is trapped during a period of well-being, and then released due to needs for protection, feeding or activities to compensate for changes in the environment. This implies a fractal Lévy walk behaviour [19].

With these two assumptions, the probability of survival (P(t)), while the system remains alive without accidents or diseases, can be expressed as:

P ( t T ) = A ( t T ) α + B (2.2)

Valid for 0 ≤ t ≤ T

According to the assumptions the power law model of Equation (2.2) follows the fractal kinetics [20] [21] with 0 < α < 1 .

Equation (2.2) denotes the state of vitality of a representative member of the species at the time t. The vitality capacity of the prototype is contained into the information of the parameters T and α. The constant B is also defined by the initial probability as B = P ( 0 ) , protection at birth.

In fact, T is characteristic of a community of species in a given environment, integrating their interaction with it, which is translated into information and is part of the system. The parameter alpha (α) represents the information of the gene pool revealed in the autopoiesis activity [22].

We make a complete formulation when the probability is normalized to one,

∫ 0 1 P ( t T ) d ( t T ) = A ∫ 0 1 ( t T ) α d ( t T ) + ∫ 0 1 B d ( t T ) = 1 (2.3)

The result is the following,

A α + 1 + B = 1 (2.4)

The constant A must be expressed in terms of the gene pool parameter (α) and the initial probability. Therefore, in the model there are three free and independent parameters, which must be determined due to the lack of information on any species (α, B and T). The probability of survival always increases except when there is an accident or a disease, in which the probability could decrease but, if there is life, the property of monotonous growth is maintained and the probability will rise to the extent of the circumstances.

The members of a species are always working to keep their survival high and preserve life (struggle for life). The search for the highest probability of survival eventually derivatives from aggressive [23] or cooperative [24] social behaviour, and the results obtained with advantage are illustrated in the first characterize of life. The dependence of probability of survival regardless the reduction of the time variable (t/T) including the genetic factor for several values of parameter (α) is shown graphically in Figure 1.

The result of the established survival model shows that, regardless of the animal species, given a pair of parameters α and B, and the normalized probability (P) of survival (representative of organic and reproductive maturity), the same life expectancy is achieved in half the time (see colour lines and inset in Figure 1). This means that the development of an animal species in a given environment has achieved the same probability of survival in the middle of its expected life, including its reproduction.

The model described above corresponds to the ideal case, in which the aging process does not occur, and death occurs accidentally. In contrast, and according to Bortz II [25], aging should be included to show that the probability of survival is reduced by the loss of physiological abilities [26]. This suggests the modification of Equation (2.2) [27] and it is proposed to add a fractal evolution term decreasing the ideal probabilities in the following way

P ( t ) = P 1 ( t ) + P 2 ( t ) (2.5)

where the first term is the probability initially proposed in Equation (2.2), still valid for 0 ≤ t ≤ T . Now, it can be called living probability.

The second term in Equation (2.5) represents the decrease in the probability of survival, or better known as the probability for death, due to aging. We assume that this starts at the critical time t 0 and it can be expressed as a negative function.

P 2 ( t ) = − D ( t − t 0 T ) β (2.6)

Valid for t 0 ≤ t ≤ T

where t₀ is the starting time of the effects of aging and β is the fractal exponent associated with damage, different from α, since the exponent β depends more on the living conditions and metabolism than on the genetic load [28]. In terms of simplicity, D = A indicates that aging has the same statistical weight as individual survival.

Particularly, equation (2.6) does not necessarily describe death as a case in which P(t) = 0 at t = T. Death is still accidental or caused by the inability to survive, the whole organism is no longer sustained.

Note that, if t₀ equals zero it would indicate that the aging process starts from birth. In this case, we could consider that a damaged organ or tissue was already present at a given time [29]. For simplicity and to separate the effects of the two processes described in Equation (2.6), the starting time is considered as

t 0 = T 2 (2.7)

Now, the survival curve would show a maximum around t₀ as it is shown in Figure 2.

It should be noted that the presence of aging does not violate the first statement, which guarantees the growth of survival probabilities. Explained in this way: the life support provided primarily by inheritance is diminished by mutation or disease.

We would say that life has been defined as the continuity in health that sustains physiological functions. The aging process [30] grows at the expense of the reduction in the number of healthy cells until it causes the break point, when it is no longer possible to keep the organism alive.

The survival growth argument in the first characteristic of life is supported by cloning [31], cryopreservation of organs and cells [32] and anhydrobiosis [33]. Stating that aging destroys the effect of life but does not eliminate the inherent vital property that biological matter possesses.

The first characteristic of life is related to the activation entropy (ΔS), defined according to the relation

P ( t ) = B e Δ S k (2.8)

where, k is a representative constant. The activation entropy (∆S) is equal to

Δ S = S − S 0 = k ln ( P ( t ) ) (2.9)

where P(t) is defined as before.

The reference of S is given at t = 0, where P ( t = 0 ) = B , then S 0 = k ln B .

The meaning of S is established in the theory of fluctuations of thermodynamics [34] and represents the entropy that must be produced in each step of the physiological processes to maintain life, in parallel to the requirements of metabolism. In this way, we can say that the activation entropy of life is associated with the increased vitality.

One of the possible conditions of biological evolution could be to maintain a steady state of minimum entropy production independent of the external environment. Sudden variations in external environments play an important role as they can alter clinical parameters and put the stability of organisms at risk. The search for a return to a stable state is possible through the internal control of organisms and efforts will be made to achieve a similar stable state.

In this sense, it is possible to consider the evolutionary transit within a development to avoid internal chaos [35] [36]. To supress that, evolution tends to establish changes in biological systems. Consequently, it is postulated that the development of organs occurred by natural selection in the search to organise physiological functions around stable internal states.

A biological system is related to its environment generating a nest that protects it against environmental changes and at the same time allows its adaptation by operating as a thermostat (regulator of vital thermodynamic parameters around clinical values or intervals). The meaning of the operation of a thermostat is taken in the broadest physiochemical sense from R. Kubo [37], who cited Bridgeman’s work [38] when he said that: “Life then appears in some way to oppose the otherwise universal drive to disorder. Not only internal but also external forces should be present to enhance energy production inside the biological systems, e.g., metabolism. Periodic oscillations of the environmental parameters around the mean values play the role of such forces. Let us stress that the essential periodically changing external factors are an inseparable thermodynamic effect of the environment on the evolution of the biological system [39]. Hence, the joint effects of internal thermodynamic factors (expressed inside the system) and external thermodynamic effects (changes and oscillations of the environmental physical parameters) determine the direction of evolution”.

We consider that the adaptation of an animal species is established when the living conditions are achieved in the proximity of a stable state with a minimum production of entropy, according to the external conditions.

“The necessary and sufficient condition for survival-adaptation is to keep physiological operations close to a state of minimum entropy production, compatible with environmental conditions”

The logical content of this statement is conditional, if a biological species achieves adaptation, it ensures that it is operating in the vicinity of a steady state of minimal metabolism according to the genotype and environment. On the contrary, if the latter is not achieved, there would be no adaptation and the species remains endanger of extinction. The underlying mechanism of adaptation is predicted to be natural selection.

We will illustrate the previous statement with the following model of evolution of the skin of homeothermic vertebrates.

The heat flow through the skin connects the transfers of thermal energy from the interior to the exterior of living organisms. In the case of homothermic animals (warm-blooded animals) such as mammals and birds, the internal temperature is maintained through the internal energy dissipation mechanisms, independent of the temperature of the environment within certain limits. For example, the internal temperature of the human being oscillates between 28˚C and 36˚C without any detrimental effect on the organic system.

A homeothermic organism has the need to transmit heat to the external environment to keep the internal temperature constant. This energy transfer makes it a dissipative structure [40]. The means of transfer are radiation (due to the difference in exterior and interior temperatures), convection (respiration) and conduction (through the skin). This section focuses on the heat loss from the organic interior through the skin. Figure 3 shows a diagram of the temperature distribution around the skin.

The skin layers (covers) of different constitutions acquired by vertebrates such as hair, feathers and shells, represent a protection against the convection of the environment around the epidermis considered as the coat or buffer of the wind chill effect.

Biological evolution has led to the establishment of an organic structure in the skin to aid in the performance of exothermic function along with changes in the metabolic rate and diet. Currently, the skin of higher vertebrates with thermal homeostasis is a system composed of three diffusive membranes, the dermis, epidermis and hypodermis, each with different thermal conductivities. The skin integrated system has an effective thermal conductivity, which will be considered as constant for all purposes, depending only on the genotype.

The skin is the thermal barrier that separates two calorific reservoirs, the first one is the internal environment at the temperature T_in and, the second is the external environment at a temperature T_ex. Now, a cross section of unit area has the same constant temperature, which changes from the inside to the outside. Heat flow is considered only in the perpendicular direction to the cross section, since the considered section can be taken as isolated, and there are no lateral flows, the temperature distribution profile is linear. The amount of heat transmitted depends on the temperature gradient, that is, the external and internal temperature difference along the thickness of the skin along the z axis. According to Fourier’s law, heat flux depends on the temperature variation in a linear relationship between the flow and the thermodynamic force, which is expressed in terms of the temperature gradient T.

q = − k d T d z (3.1)

where k is the effective thermal conductivity, a parameter considered constant for the skin of a species in the time of exposure to the environment considering an experiment of a certain duration.

With these assumptions, the above flow-force relationship gives rise to steady state of minimum entropy production, and the steady state is stable against fluctuations. Conclusions are detailed below.

It is established that the entropy production (σ) expressed by the product of the heat flow and the thermodynamic force is given by

σ s = q d T d z > 0 (3.2)

The total production of entropy (P) in the system (Lyapunov’s function of a dynamic system) is the integral in the entire volume (U) surrounded by skin,

P = ∫ σ d U = ∫ q d T d z d U (3.3)

The minimization of this function is achieved by varying the total entropy production (fluctuations of internal or external origin) and equalling to zero, which leads to

d q d z = 0 (3.4)

Which putting in the energy balance equation leads to the following result [41]

∂ T ∂ t = 0 (3.5)

Assuming there is a stationary state, in which the profile of internal and environmental temperature does not change. This shows that the minimum entropy production is associated with a steady state.

Following with the second part of the proof of principle of least entropy, the stability of the steady state will be demonstrated. See the verification of the minimum energy principle in the Appendix A.

∂ P ∂ t = − 2 ∫ ρ c v T 2 ( ∂ T ∂ t ) 2 d U < 0 (3.6)

Equation (3.6), shows the entropy production decreasing with time with a vari- ation of the state, verifying the stability of a steady state with minimum entropy and minimum metabolism [42]. Note that the quantities in the integrand are all positive, which results in the variation in entropy production decreases with time. In terms of this theorem established that a thermodynamic system in a steady state is stable with a minimum entropy production.

Now, we will illustrate how in the presence of external convection the validity of this principle is maintained by lowering the temperature, but its breaks down before freezing,

Consider Fourier’s law applied to the skin in the absence of a convective effect (see curve (a) in Figure 4) the heat flux from the interior to the environment is established by

q 1 = k e f L 0 ( T i n − T e x ) (3.7)

In the presence of the wind chill effect, the internally produced heat flow increases when the external temperature decreases as a function of the wind speed V, observing an equivalent external temperature T e x ( V ) , where the heat flow (see curve (b) in Figure 4) is expressed as

q 2 = k e f L 0 ( T i n − T e x ( V ) ) (3.8)

q 2 > q 1 (3.9)

On the other hand, a linear temperature profile is maintained throughout the skin due to the generation of a dynamic change introduced by apparently reducing the thickness of the skin.

The following model exemplifies how the thickness of the skin is reduced, as we will see below. Considering the same amount of heat that is produced to maintain the internal temperature when observing that the initial external temperature ( T e x ) has penetrated to a certain depth due to the effect of the wind,

q 2 = k e f L ( T i n − T e x ) (3.10)

The effective thickness of the skin is L and not L₀. That is, the difference in the thickness of the skin stops working to counteract the cooling.

By matching the two last Equations (3.8) and (3.10) the effective thickness of the skin operating in the presence of the wind chill effect is given by L (see dotted line (c) in Figure 4) is

L = L 0 ( 1 − x ) (3.11)

where

x = T e x − T e x ( V ) T i n − T e x ( V ) (3.12)

here is a closer look at the limit cases of x. First, it is zero when the wind speed is zero and T e x = T e x ( V = 0 ) . Second, when x → 1 , this limit is not reached, since a decreased in internal temperature occurs due to the arrival of white noise (chaotic behaviour), and freezing can occur, causing the situation that T i n → T e x . For illustration purposes, consider T e x = 10 ˚ C , wind at 40 km/h, and the equivalent temperature (wind chill [43]) is the freezing [44] (frostbite) one equal to T f r o s = − 27 ˚ C . Using the values in Equation (3.12) for Kelvin temperatures, it gives x = 0.58. In this example we can see that since the freezing temperature was reached, our model is not valid because the skin is no longer a thermal barrier.

We now consider T e x ( V ) as the one established in the Wind Chill Temperature (WCT) chart. This chart, with air temperature in ˚C and wind speed in km/h, expresses the equivalent temperature with the following equation [45]

T e x ( V ) = 13.12 + 0.6215 T − 11.37 V 0.16 + 0.3965 T V 0.16 (3.13)

In this Equation (3.13) the three effects of heat transmission, conduction, con- vection and radiation, have been considered.

The relation of L and L₀ according to the Equations (2.11) and (2.12) is given by

L L 0 = T i n − T e x T i n − T e x ( V ) (3.14)

Figure 4 shows the variables in the considered system. The layers of the skin are shown and how the dermis (skin) is protected by the epidermis and hypodermis.

Figure 5 shows L/L₀ versus the wind speed (V) within the interval in which the organism manages to recover its normal internal temperature after cooling, by increasing the wind chill effect.

Figure 5 shows the relationship of the effective thickness of the skin in the presence of the wind chill effect with respect to the original thickness. This figure shows that the external temperature can penetrate up to 20% into the skin and the profile decreases. According to the reference of the effects of the convection of the wind against the skin, exposure to winds greater than 40 km/hour at a temperature of −150˚C produces, according to our model, a penetration into the skin of the external temperature at depths greater than 20% is life-threatening.

In our model it is shown that in the presence of the wind the thickness of the skin decreases to a value L [see Equation (3)]. An important appreciation at this point is that the external temperature and the thickness of the skin change, and the variation L specifies the thermal sensation of cold. This is accomplished using equivalent temperature table Figure 6.

The behaviour of the dividing line between before and after freezing, depends on V and T_ex. Linear regression offers the following equation

L s L o = A T e x ( V ) + B (3.15)

where, A = −0.0162 and B = 0.5639

Figure 7 qualitatively shows the change from the linear path to the non-linear, established by the limit L s .

Figure 7 shows the effective length (L_s) of the skin at the freezing point (see Equation (3.15). It is assumed that white noise occurs when the heat transport equation becomes stochastic [46], when L changes for L₀ (L < L₀).

The limit of the change in L remains at the freezing point denominated by L_s. According to the model presented, L varies between the following limits.

L s < L < L 0 (3.16)

Given that non-linearity (white noise) is accompanied by a state of stress, the evolution of the skin has been guided to protect it from the “wind cooling effect”, achieving a linear temperature profile in the skin and stability against changes of temperature. This stability of the biological system has been integrated into physiological regulation by evolution. In other words, the evolution of biological systems has found a solution and can be exemplified in our model to show the effect of convection.

Based on this result, it is proposed body coverings of different constitutions that vertebrates have acquired [47], such as feathers and shells are explained by the search for an evolutionary trend that mitigates the effect of the wind cooling effect and brings the thermal regulation of homeothermic animals closer to the state of minimum entropy production.

E. Schrödinger in 1944 [61] proposed the term “Negentropy” to indicate the property of biological matter to oppose a tendency to increase disorder and entropy, as stated in the second principle of thermodynamics. Instead, biological matter self-organises in the form of aperiodic structures with the possibility of genetically encoding the acquired structure in the DNA molecules [62].

Thermodynamically, the evolution of living matter is described by postulating the validity of the negentropy principle [63], which states that the evolution of biological system occurs in the direction [64] which the structure [65] becomes more complex. That is, the phylogenetic changes occur in the direction in which the negentropy of the structure increases.

In a biological system, the entropy after a series of evolutionary steps is S₁, and the change (∆S₁) considering the reference of a totally disordered system with the same molecules (S₀) is given as

Δ S 1 = S 1 − S 0 (4.1)

Suppose that the same system (S₁) continues to evolve, it will meet S_t, the entropy change regarding the previous state is

Δ S t = S t − S 1 (4.2)

Negentropy is defined as the decrease in entropy that has accumulated over the course of successive generations.

Δ H = − ( Δ S t − Δ S 1 ) = S 0 − S t > 0 (4.3)

In the total evolution, a total change in negentropy has been gained, given by,

Δ H > 0 (4.4)

The ∆H is the negentropy intake (see Figure 8) and represents the gain in structure for a biological specie through its life history on earth that has taken about 3465 million of years [66].

Figure 8 shows the maximum entropy of a biological system, which depends on time according to its own morphogenetic evolution, recognised as phenotypic plasticity [67]. We take S₀ as a reference for the maximum entropy established by the total molecules that make up the biological body in the form of a simple mixture (homogenous system). As an example, a random set of molecules (S₀) is taken. The entropy of a biologically structured (heterogeneous) system is (S₁) which is a quantity that continues to decrease over time (S₂) and so on (S_f), (see dashed lines), and the total change is Δ H = S 0 − S 2 > 0 .

Then S₁ is that entropy when the system has gained a certain structuring of the molecules to conserve and extend their life. That is, the dashed curve of S₁ is the survival line of a heterogeneous biological system and similarly, the dashed curve of S₂ and so on S_f. The continuous curve is the survival path of a heterogeneous biological system in a post evolution scenario.

From a thermodynamic point of view, genetic mutations are modelled representing a thermal fluctuation creating a genetic constriction. When a constraint is removed, the system spontaneously changes as in a free expansion (constant internal energy) and the entropy increases (dS > 0). The opposite process establishes a constriction where the entropy decreases by which a mutation is established.

Consider a reversible process of a microsystem in contact with a heat reservoir at a constant temperature T. An external disturbance (could be a fluctuation) acts ( δ W r e v ) on the microsystem. In this case, the entropy change is negative (dS < 0), given by the equation

δ W r e v = − T d S = − δ Q (4.5)

The term to the right of the equalities (rhs) is the amount of heat ( δ Q < 0 ) transferred to the surroundings so that the internal energy does not change. The total evolutionary work of a species is the repeated sum of the accumulated changes

W r e v = − ∫ T d S = − T Δ (4.6)

On the other hand, negentropy (∆H) is given in terms of all accumulated works as

Δ H = W r e v T (4.7)

and represents the degree of organization [68] established in the biological species.

The constant increase in this quantity (H) indicates that the macroscopic process is biologically irreversible as the increase in the macroscopic and microscopic organization of DNA.

In terms of predictability, future events have several forms of occurring and depend on many fluctuations. However, past events are recorded and could be genetically determined, curiously in the direction in which negentropy increases. This can be seen in the validity of the description of the molecular clock hypothesis [69], phylogenetic trees [70] and evolutionary ladders [71].