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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Coherent states of Gompertzian growth

Marcin Molski, Jerzy Konarski

Published Aug 25, 2003• 44 citations
Publisher

Abstract

The origin of the Gompertz function G(t)=G(0)e(b/a(1-e(-at))) widely applied to fit the biological and medical data, particularly growth of organisms, organs, and tumors is analyzed. It is shown that this function is a solution of a time-dependent counterpart of the Schrödinger equation for the Morse oscillator with anharmonicity constant equal to 1. The coherent states of the Gompertzian systems, which minimize the time-energy uncertainty relation, have been found. These are eigenstates of the annihilation operator identified with the operator of growth, whereas eigenstates of the creation operator represent the Gompertzian states of regression. The coherent formation of the specific growth patterns in the Gompertzian systems appears as a result of the nonlocal long-range cooperation between the microlevel (the individual cell) and the macrolevel (the system as a whole).

DOI
10.1103/physreve.68.021916

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The origin of the Gompertz function G(t)=G(0)e(b/a(1-e(-at))) widely applied to fit the biological and medical data, particularly growth of organisms, organs, and tumors is analyzed. It is shown that this function is a solution of a time-dependent counterpart of the Schrödinger equation for the Morse oscillator with anharmonicity constant equal to 1. The coherent states of the Gompertzian systems, which minimize the time-energy uncertainty relation, have been found. These are eigenstates of the annihilation operator identified with the operator of growth, whereas eigenstates of the creation operator represent the Gompertzian states of regression. The coherent formation of the specific growth patterns in the Gompertzian systems appears as a result of the nonlocal long-range cooperation between the microlevel (the individual cell) and the macrolevel (the system as a whole).
A model of living systems considered as coherent, time-dependent anharmonic oscillators is presented. It is based on the concept of space-like coherent states minimizing the time-energy uncertainty relation, adapted to the case of biological systems whose growth is described by the Gompertz or West-Brown-Enquist functions. The coherent states of biological growth evolve coherently in space being localized along the classical time trajectory; hence, the growth is predicted to be coherent in space. It is proven that the Gompertz function is a special solution of the space-like Horodecki-Feinberg equation for the time-dependent Morse oscillator in the dissociation state. Its eigenvalue represents the momentum of biological growth, associated with a space-like component whose properties resemble those attributed by vitalists to the life momentum or vital impulse. The physical characteristics of the life energy and momentum and their connection with the concept of zero-point momentum of vacuum are presented.
The phenomenological universalities (PU) are extended to include time-depended quantum oscillatory phenomena, coherence and supersymmetry. It will be proved that this approach generates minimum uncertainty coherent states of time-dependent oscillators, which in the dissociation (classical) limit reduce to the functions describing growth (regression) of the systems evolving over time. The results obtained reveal existence of a new class of macroscopic quantum (or quasi-quantum) phenomena, which may play a vital role in coherent formation of the specific growth patterns in complex systems.
Based on Gaussian wave packet solutions of the time-dependent Schrödinger equation, a generalization of the conventional creation and annihilation operators and the corresponding coherent states can be obtained. This generalization includes systems where also the width of the coherent states is time-dependent as they occur for harmonic oscillators with time-dependent frequency or systems in contact with a dissipative environment. The key point is the replacement of the frequency ω 0 that occurs in the usual definition of the creation/annihilation operator by a complex time-dependent function that fulfils a nonlinear Riccati equation. This equation and its solutions depend on the system under consideration and on the (complex) initial conditions. Formal similarities also exist with supersymmetric quantum mechanics. The generalized creation and annihilation operators also allow to construct exact analytic solutions of the free motion Schrödinger equation in terms of Hermite polynomials with time-dependent variable.
During the century of existence of the notion of coherent states, either linear or nonlinear, several schemes for their construction, theoretical or experimental, have been developed. Generally, the mathematical structure of coherent states depends on the choice of ladder operators, and consequently on the structure constants. In this paper, we propose a way to construct generalized coherent states for anharmonic oscillators that is based on a diagonal operator ordering technique (DOOT) applied to generalized hypergeometric functions, that is, on some of the most general special functions. These states are generated by the action of a pair of ladder operators, the creation and the annihilation, whose ordered normal product is equal to the dimensionless Hamiltonian of the quantum system. In addition, the action of these operators is easy to find if the expression for the dimensionless energy eigenvalues is known.
This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.