# Far from Equilibrium Percolation, Stochastic and Shape Resonances in the Physics of Life

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Percolation: The Careri Papers

_{c}—separating two regions: one with very low and constant capacity and the other characterized by a rapidly growing capacity, respectively below and beyond the threshold. The numerical value of h

_{c}was remarkably stable with respect to variations of the chemical properties of the system, like the pH or the chemical nature of the solvent. They interpreted these results as a signature of critical behavior: the dramatic change in the electrical properties was due to the sudden appearance of long-range correlations in the form of a macroscopic network (i.e., ranging over the size of the whole sample) of microscopic conducting elements. The quantitative analysis of this threshold showed a percolation of protons along threads of hydrogen-bonded water molecules on the surface of lysozyme proteins.

#### 1.2. Criticality

## 2. Percolation: Mathematical Description

_{c}is approached, a cluster whose dimensions are comparable to the system size appears.

_{s}the probability that a site is part of a cluster of size s. This probability gives us a great deal of information about the structure of the system. It can be easily guessed, from inspection of limiting cases (like for example the one-dimensional case in which the lattice reduces to a linear chain), that for large s this probability is dominated by an exponential decay m

_{s}∝ e

^{−}

^{cs}where c is a function of p. In order to proceed further and to find the correct functions that describe the dependency of m

_{s}on p an assumption that takes the name of scaling hypothesis has to be made:

^{z}f (z)= e

^{z}(with τ = 2, σ = 1 ) and the infinite-dimensional case (realized on a particular topology called the Bethe lattice).

## 3. Stochastic Resonance

#### 3.1. Biological Systems and the Benefits of Noise

#### 3.2. Brief Description of the Mathematical Model

_{t}is a normalized white noise (i.e., with zero mean and unitary standard deviation) and ɛ is a parameter setting the amplitude of the fluctuational force. In the case of a particle immersed in a fluid we have ɛ = 2D where D is the temperature-dependent diffusion coefficient. The fluctuational forces cause interwell transitions with a rate given by Kramers reaction-rate theory [23]:

_{0}cos(Ωt) the double-well symmetric potential is tilted asymmetrically up and down, and in each instant of time the potential has two wells but with a different height. The hopping between the two periodically changing wells can then be synchronized with the weak periodic forcing. This synchronization takes place when the noise level is tuned at the value in which the typical waiting time between the transitions is comparable with half the period of the periodic forcing.

#### 3.3. Stochastic Resonance on Network Structures and in Biological Systems

## 4. Shape Resonance

#### 4.1. A Brief History of the Shape Resonance

_{2}→H

_{3}→H +H +H that shows the key quantum mechanical feature of the configuration interaction between a bound state and a continuum described by the shape resonance (see Figure 5). The quantum interferences lead to exciting situations where long-lived metastable states (quasi-stationary states) coexist with ultrafast processes observed experimentally. This phenomenon is usually called the “trapping effect” or “avoided resonance overlapping” and has been studied in numerous fields of physics and chemistry [72–74] driven by the dynamics of a state resulting from the superposition of a discrete state with a continuum one.

_{c}. The complexity and shape resonances appear to be the key for keeping quantum coherence at high temperature. The time evolution of this complexity is now of high interest since its control will allow the development of advanced material with novel functionalities [89]. The complex structural multiscale phase separation of these systems appears similar to structural features of living matter [90].

#### 4.2. A Brief Description of the Shape Resonance through Simple Examples

^{2})+hν →He(1s

^{0}2s

^{1}np

^{1}) with mixing between quasi boud stationary state with continuum states He(1s

^{0}2s

^{1}np

^{1})↔He(1s

^{1})+ɛp, where ɛp is a free photoelectron, by Ugo Fano [65,66] who obtained the formula for the spectral line shape of the scattering cross-section:

_{0})/Γ where E

_{0}is the resonant energy and Γ is the width of the autoionized state, this equation predicts a maximum and a minimum in the Fano line-shape. The Fano formula is a superposition of the Lorentzian line shape of the discrete level $\frac{{q}^{2}-1}{{\varepsilon}^{2}+1}$ with a flat continuous background and a mixing term $q\frac{2\varepsilon}{{\varepsilon}^{2}+1}$ The formula 6 describes well the experimental atomic photoabsorption cross-section for the excitations of two electrons above the continuum threshold or the cross-section for the capture of a free neutron captured into the nucleus around the energy of the nuclear shape resonance. The deviation of the cross-section from the Lorentzian is measured by the Fano-line-shape parameter q. The value of q can be easily extracted form experiments. For a very large value of q the line shape approaches a Lorentzian because of the lack of quantum interference between the states in the continuum and the quasi-bound states as shown in Figure 6. By decreasing the value of q a minimum develops at energies lower than the energy of the quasi bound state at E

_{0}and the maximum moves well above the quasi-bound state. In the extreme case of strong interference, proportional to 1/q, the minimum of the cross-section moves close to the quasi bound state energy E

_{0}and becomes very large while the maximum moves far away from E

_{0}and its intensity vanishes. The result is that for very high interference the shape resonance gives essentially an anti-resonance driven by the negative interference effects.

_{1}and ω

_{2}connected by a spring. If some energy is injected into one of the oscillators at t = 0 it will then flow into the second one, while the first will go back to the resting state. After a while the energy from the second oscillator will go back to the first oscillator and so on. The resonant exchange becomes larger as the frequency difference ω

_{1}−ω

_{2}→ 0. In quantum physics the analogue case is the configuration interaction between two bound states confined in the same atom or in the nucleus and it depends on the energy difference between the levels.

_{12}describes the strength of the coupling between the two oscillators. Solving this set of equations and calculating the amplitudes, one finds that the resonant behavior of the amplitude as a function of the frequency of the external force has two peaks (symmetric and asymmetric line-shape) near the two “eigen” frequencies. The coupled oscillator responds with two resonances 1) a symmetric resonant profile at ω

_{1}and an asymmetric profile at ω

_{2}. The first is described by a Lorentzian function (known as a Breit-Wigner resonance) and the second is characterized by the asymmetric Fano line-shape as an effect of the phase interference of the two oscillators when driving frequency passes through the resonance.

_{n}at site n and nearest-neighbor coupling with strength C. This system supports propagation of plane waves with dispersion ω

_{k}= 2C cos k. The second subsystem consists of a single state ψ with energy E

_{b}. The third term the coupling among the chain and the oscillator given by the coupling coefficient J between the state ψ and the state φ

_{0}of the discrete chain. A propagating wave may directly pass through the chain or instead visit the Fano state, return back, and continue with propagation. The main resulting action of the Fano state is that the strength of the effective scattering potential J

^{2}/(ω

_{k}− E

_{b})resonantly depends on the frequency of the incoming wave ω

_{k}. If E

_{b}lies inside the propagation band of the linear chain |E

_{b}| < 2C, the scattering potential will become infinitely large for ω

_{b}= E

_{b}, completely blocking propagation as shown by the strong anti-resonance with a dip in Fig. 6 at high strength of the mixing term. Therefore meeting the resonance condition, leads to a resonant suppression of the transmission, which is the main feature of the Fano resonance.

_{c}and operators b

^{+}and b. This state is usually called impurity. The second term describes a continuous set of states with energy ɛ

_{κ}and k wave-vector and operators c

^{+}and c that have a finite band-width given by the tight-binding models of solids or by a free-electron model. The third term describes the mixing between these two kind of states describing the process where the continuum particle hops onto the impurity (b

^{+}c

_{k}) or particle hops off the impurity into the continuum (c

_{k}

^{+}b) that describe the scattering resonance for the particle at the Fermi level. The particles can be bosons or fermions, in this last case they preserve the spin orientation jumping from the bound state to the continuum and vice-versa.

_{kb}and ɛ

_{kc}of the two bands and c, c*, b and b* are the corresponding annihilation and creators operators. The third and fourth terms V

_{cc}, V

_{bb}, are the weak electron-electron attraction forming the cooper pairs and giving the BCS condensates in each band. The summations extend only over different k⃗ values in the two bands corresponding to energies with a distance ± Ħ ω of the Fermi surface. The fifth and sixth terms are the Fano-like mixing terms due to mixing of the two condensates with the transfer of a pair from the b to c band and vice-versa.

_{k,k}

_{′}J (k,k′) (b

^{*}

_{k}

_{↑}b

^{*}

_{−}

_{k}

_{↓}c

_{−}

_{k}

_{′↓}c

_{k}

_{′↑}+c.c.) where the exchange integral J (k, k′) may be repulsive or attractive is responsible for the increase of the transition temperature in complex multi-Fermi surfaces giving multigap superconductors [81,87] also in the case where pairs are formed only in one of the two band. In the extreme case of a first bosonic set of states (hard core bosons “b”) and itinerant electrons “c” the Hamiltonian is given by [97]

#### 4.3. Shape Resonance in Biology

_{d}and is the inverse of the association constant, that in the special case of salts can also be called an ionization constant. For a general reaction:

_{x}B

_{y}breaks down into x A subunits and y B subunits, the dissociation constant is defined as

_{x}B

_{y}] are the concentrations of A, B, and the complex A

_{x}B

_{y}, respectively. At each time step new complexes can form and existing complexes can break down into smaller complexes according to experimental interaction rules provided as input to the model.

## 5. Conclusion

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**Figure 1.**(

**a**) Definition of percolation and clusters. Some squares are occupied by big dots and some are empty. The clusters are groups of neighboring occupied sites; we filled the dots inside a cluster with the same color. (

**b**) Matlab simulation of two-dimensional (site) percolation at the percolation threshold (P

_{c}= 0.5927) on a square lattice of size L = 10

^{3}. There is a cluster (the dark blue one) spanning the whole system from the bottom to the top of the square.

**Figure 2.**The mechanism of stochastic resonance. The double well potential V

_{(x)}represents a physical system with two stable states. In absence of noise a particle, which at one time is within one of the two wells, relaxes towards the intrawell minimum and then stops. When stochastic perturbation is added to this system the particle can hop between the two wells with a rate predicted by reaction-rate theory. If a weak periodic forcing is present the overall potential minimum changes periodically from one side to the other. Stochastic resonance happens when the typical escape time from the lower barrier and the periodic forcing are synchronized.

**Figure 3.**The typical shape of a stochastic resonance peak. The behavior of the amplitude of the system response <x> is plotted as a function of the noise level given by the diffusion coefficient D.

**Figure 5.**The “shape resonance” is determined by the quantum configuration interaction between the quasi-bound states forming the bound molecules in the potential valley and the dissociated free molecules moving in the regions of the potential plain. The shape resonance modifies the wave function of the free particles and it controls the interaction between the free molecules.

**Figure 6.**The evolution of the Fano line-shape for the shape resonance, as a function of the strength of the interference term proportional to 1/q, where q is the parameter of the Fano formula.

**Figure 7.**(Upper panel): A pair of oscillators ω

_{1}and ω

_{2}is weakly linked. The energy injected into one oscillator flows to the other one and vice versa both in the classical world and in the quantum world. (Lower panel): A chain of oscillators has a weak link with a local oscillator. The energy given to the local oscillator flows towards the chain of oscillators, moves away as a wave sin(kx + ωt) and does not come back to the local oscillator in classical physics. There is no resonance in classical physics between the localized mode and the free wave. On the contrary, in the quantum world the shape resonance describes the resonance between the free wave and the local mode.

© 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Poccia, N.; Ansuini, A.; Bianconi, A.
Far from Equilibrium Percolation, Stochastic and Shape Resonances in the Physics of Life. *Int. J. Mol. Sci.* **2011**, *12*, 6810-6833.
https://doi.org/10.3390/ijms12106810

**AMA Style**

Poccia N, Ansuini A, Bianconi A.
Far from Equilibrium Percolation, Stochastic and Shape Resonances in the Physics of Life. *International Journal of Molecular Sciences*. 2011; 12(10):6810-6833.
https://doi.org/10.3390/ijms12106810

**Chicago/Turabian Style**

Poccia, Nicola, Alessio Ansuini, and Antonio Bianconi.
2011. "Far from Equilibrium Percolation, Stochastic and Shape Resonances in the Physics of Life" *International Journal of Molecular Sciences* 12, no. 10: 6810-6833.
https://doi.org/10.3390/ijms12106810