Unified description of thermal behaviors by macroscopic growth laws

Complex systems, in many different scientific sectors, show coarse-grain properties with simple growth laws with respect to fundamental microscopic algorithms. The known classification schemes of the growth laws refer to time evolution of biological and technical systems. We propose to apply the previous classifications to phenomenological analysis of thermal systems with a cross-fertilization among different sectors. As an example, the Fermi–Dirac distribution function and the electrical activation in implanted silicon carbide are discussed.


Introduction
Simulations of complex systems with a large number of interacting elementary parts are often difficult [1] and involve a large number of free parameters.
On the other hand, there is an impressive number of experimental verifications, in many different scientific sectors, that coarse-grain properties of systems, with simple laws with respect to fundamental microscopic algorithms, emerge at different levels of magnification.
In this respect, a useful tool is the general classification of growth laws in [2] which facilitates the cross fertilization among different fields of research. For example, the Gompertz law(GL) [3] applies to human mortality tables (i.e. aging) and tumor growth [4][5][6].
In general, a macroscopic growth problem is characterized by a function f (t), which describes the time evolution of some dynamical quantity, and by the specific rate, α, defined as (1/f )(df/dt)=α(t). In the GL α has an exponential dependence on time: where a and b are constants. In aging f (t) indicates the survival probability, while with regards to tumor growth it corresponds to the number of cells N(t) (depending on the specific case a and b can be positive or negative). For technical devices the specific rate of the survival probability has a power-law time behavior with n>1, called Weibull law (WL) [7,8].
In the previous equations t is the time variable. However, formally, it can represent any parameter useful for describing the system.
We discuss the application of the growth laws to thermal behaviors, i.e. to the temperature, T, dependence of dynamical quantities. The interpretation of the parameter t as 1/T (with the Boltzmann constant k B =1) reproduces some typical laws of thermal systems and, as an example, the formalism will be applied to the Fermi-Dirac (FD) distribution function and to the electrical activation of the implanted silicon carbide (SiC) after annealing.
In section 2 we recall the classification of the growth laws; in section 3 one considers the generalization to thermal behaviors and the applications to FD distribution and to SiC; section 4 is devoted to comments and conclusions. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Classification of growth laws
Many different biological systems evolve according to the simple equation where α is the specific growth rate. A classification of the previous growth laws is obtained by considering the power expansion in α of the function (see [2] for details) gives a t-independent specific rate α 0 and therefore an exponential growth; for b 0 0 ¹ and b i =0 for i>1 describes a linear t-dependent specific rate and again an exponential growth; at the first order in α, for b b 0, 0 0 1 = ¹ and b i =0 for i>1, reproduces an exponential behavior in t of the specific growth and therefore the GL; the second order term, generates the logistic and generalized logistic growth for f (t).
The feedback effect, that is the dependence of the specific growth rate α on the function f (t), can be easily derived by the dependence on t of the specific rate. For example, the GL for a growing number of cells, N(t), shows the well known logarithmic non linearity, and for the (generalized) logistic law one gets the typical power-law behavior where a b c , , , g are constants and the carrying capacity, N ¥ , corresponds to fixed point where α=0. For technical devices, the previous classification scheme has been generalized since the specific growth rate of Weibull law has a power dependence on t which is not reproduced by equation (4). The behavior α f (t) ; t n , with n positive integer, corresponds to terms O ( n n 1 a -) ( ) in the expansion of Φ(α) and therefore for a general classification scheme of the specific growth/aging/failure rate of biological and technical systems one has to consider [9]: Note that: (a) 0<(n−1)/n<1 and the n-th term in the power series in α ( n−1)/ n tends for large n to α, i.e. to the Gompertz law; (b) the term b 0 0 ¹ , i.e. the exponential growth, can be neglected if one considers the GL, the generalized logistic or more complex growth laws for the biological systems (there is no problem to include this term in the expansion); (c) the first sum in the expansion has fractional powers. As a by-product of the proposed classification scheme one can easily evaluate the aging/failure of combined new bio-technical 'manufactured products' [9].
On the other hand, in many different systems it is often studied the dependence of the dynamical variables on the temperature T and one wonders about the role of 'growth' laws in such cases. In other terms the question concerns the information contained in equations (3), (4) if, for example, one interprets the variable t as 1/T (with the Boltzmann constant k B =1), i.e. by considering the equations: In the next section we shall analyze this aspect.

Thermal behavior
Let us now study some specific solutions of equations (8), (9)  logistic) equation for f, that we discuss in more details in the following applications.

Fermi-Dirac distribution
The Fermi-Dirac distribution for a free fermion gas is given by where Δ is the difference in energy with respect to the Fermi energy. At fixed Δ, as a function of 1/T,f satisfies a logistic equation. Indeed, According to the classification in [2], the logistic feedback in equation (11) corresponds to the class U2 [2], i.e. it is obtained by the expansion of Φ(α) in equation (9) to second order in α, as one can easily verify by the chain of relations i.e. c 1 =Δ and c 2 =1 in equation (9).
The fixed point f FD =1 is due to the Pauli principle and indeed the Bose-Einstein distribution, where E is the energy, follows the T evolution equation The classification U2 implies that two physical constants completely specify the macroscopic evolution rate in T, independently on the microscopic details. A more complex system, in a complete different sector, will be discussed in the next subsection.

Silicon Carbide electrical activation
Silicon Carbide is a wide bandgap semiconductor with high thermal conductivity and other outstanding properties [11,12] used in a large series of electronic devices. An accurate, microscopic, modeling of its electrical properties is currently not available and phenomenological approaches are often proposed [13]. The previous 'thermal growth' classification immediately suggests various empirical models of the electrical activation as a function of the annealing temperature at fixed implantation dose. The first model is obtained by considering equation (8) for the electrical activation, I, and by considering the first order expansion Φ(α)=c 1 α , which corresponds to the GL, i.e.
where k g is a constant and I 0 =1, i.e. 100 % of possible activation of the implanted ion, for low implantation dose ( 10 cm 14 2  -).
The second and third empirical models are obtained by considering the expansion of Φ(α)=c 1 α+c 2 α 2 which generates the logistic behavior, The previous analysis can be easily generalized to take into account that, at large implantation dose ( 10 cm 14 2 > -), high temperature thermal processes reduce the maximum electrical activation.  which implies that there is a typical time scale of the process, related to α, and that the system has a carrying capacity, i.e. a value f k 1 * = where there is no growth, i.e the specific rate has a fixed point. These two dynamical ingredients are common to many different systems, of course with different meanings.
In cellular growth there is a characteristic time due to cell doubling and a carrying capacity related to the available metabolic resources [20].
The carrying capacity (i.e. the saturation point) for SiC electrical activation is essentially due to the presence of defects and to solid solubility: there is a maximum number of defects that can contribute to the electrical activation.