Deceased on 13 October 2012.

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We consider the case of a peculiar complex behavior in open boson systems sufficiently away from equilibrium, having relevance in the functioning of information-processing biological and condensed matter systems. This is the so-called Fröhlich-Bose-Einstein condensation, a self-organizing-synergetic dissipative structure, a phenomenon apparently working in biological processes and present in several cases of systems of boson-like quasi-particles in condensed inorganic matter. Emphasis is centered on the quantum-mechanical-statistical irreversible thermodynamics of these open systems, and the informational characteristics of the phenomena.

More than forty years have elapsed since the renowned late Herbert Fröhlich first presented his concept of long-range coherence in biological systems [

“... under appropriate conditions a phenomenon quite similar to a Bose condensation may occur in substances which possess longitudinal electric modes. If energy is fed into these modes and thence transferred to other degrees of freedom of the substance, then a stationary state will be reached in which the energy content of the electric modes is larger than in the thermal equilibrium. This excess energy is found to be channelled into a single mode—exactly as in Bose condensation—provided the energy supply exceeds a critical value. Under these circumstances a random supply of energy is thus not completely thermalized but partly used in maintaining a coherent electric wave in the substance." [

This Bose(like) condensation does not follow in equilibrium but in non-equilibrium conditions, displaying a complex behaviour consisting in the emergence of a dissipative structure in Prigogine’s sense [

In Fröhlich model vibrational-polar modes are excited by a continuous supply of energy pumped by an external source, while these modes interact with the surrounding medium acting as a thermal bath. The interplay of these two effects—pumping of energy subtracting entropy from the system and dissipative internal effects adding entropy to the system—may lead to the emergence of complex behaviour in the system consisting in what can be called

Fröhlich’s synchronous large-scale collective oscillations imply inter-cellular microwave emissions which would constitute a non-chemical and non-thermal interaction between cells. These oscillations could therefore be revealed by detection of emissions of GHz or THz radiation. Such electromagnetic signals are of extremely low magnitude and the receiver technology to measure them was not available during Fröhlich’s time. It is only now that the predicted signals can be detected by adapting technology that has been developed for space and astrophysical research. Hence, a whole new area of biology is now ready for investigation.

Earlier experiments looking after Fröhlich effect were not conclusive, but now as notice above a “second generation” of experiments are becoming available. They require further improvement, but already some preliminary results are encouraging [

Non-biological implications of Fröhlich effect could also be far-reaching. It can be mentioned some connection with homoeopathy and atmospheric aerosol physics. Regarding the latter, sunlight-pumped Fröhlich-like coherent excitations may play a role in producing anomalies in the spectrum of light absorption [

Other example where Fröhlich’s condensation and Davydov’s soliton appear to be present is the case of the so-called “excitoner”, meaning stimulated coherent emission of excitons created by random excitations, in a situation similar to the case of photons in a laser [

In conclusion, we are facing a stimulating revival of Fröhlich effect, after a certain period of partial hibernation. This revival is a strong one in the sense that, as noticed, it may open a whole and relevant new area of research in basic biology and also in the realms of condensed matter physics. Let us consider the case of biological systems.

What is biophysics? For us, life is the most important phenomenon in Nature. It is also very complex, and in order to understand life and living processes several branches of science are needed. Biophysics uses biological and physical concepts for the study of life. One of the greatest physicist of the twentieth century, Erwin Schrödinger, wrote a beautiful little book [

An article in Science [

Which may be the theoretical approach in physics to carry on a program to deal with at the microscopic as well as, at the same time, the all important, macroscopic levels of bio-systems and their synergetic aspects? During the last decades this question concerning the theoretical description of the macroscopic behaviour of dissipative open many-body systems in arbitrarily far-from-equilibrium conditions has been encompassed in a seemingly powerful, concise, and elegant formalism, established on sound basic principles. This is a non-equilibrium statistical ensemble formalism, accompanied with a nonlinear quantum kinetic theory, a response function theory for systems arbitrarily away from equilibrium, a statistical thermodynamics for dissipative systems, and a higher-order generalized hydrodynamics. This is the formalism used for the study of complex behaviour in biological systems, mainly the so-called Fröhlich’s effect and some other accompanying phenomena, as the long-distance propagation of nearly undamped and undistorted signals.

Here we present a description of these ideas applied to the study of a general case of complex behaviour in open boson systems, be it in bio-systems or in condensed matter like semiconductors. Hence, and in conclusion of this section, we may state that the results to be described, resulting from a promising particularly successful marriage of nonlinear non-equilibrium statistical thermodynamics and biology, lead us to paraphrase Herbert Fröhlich saying that it is particularly auspicious to see that biological systems may display complex behaviour describable in terms of appropriate physical concepts.

Particular complex behaviour has been observed in the case of boson systems, as Bose-Einstein condensation (BEC) in fluids in equilibrium at very low temperature. A case is superfluidity in liquid helium evidenced by Pyotr Kapitza [

A second type of BEC is the one of boson-like quasi-particles, that is, those associated to elementary excitations in solids (e.g., phonons, excitons, hybrid excitations,

The third type is the case of boson-like quasi-particles (associated to elementary excitations in solids) which are driven out of equilibrium by external perturbative sources. D. Snoke [

Several cases can be listed:

A first case was evidenced by

A second case is the one of acoustic vibration (ac phonons) in biological fluids, involving nonlinear anharmonic interactions and in the presence of pumping sonic waves, with eventual STI relevance in supersonic treatments and imaging in medicine [

A third one is that of excitons (electron-hole pairs in semiconductors) interacting with the lattice vibrations and under the action of RF-electromagnetic fields; on an STI aspect, the phenomenon has been considered for allowing a possible exciton-laser in the THz frequency range called “Excitoner” [

A fourth one is the case of magnons [

There exist two other cases of NEBEC but where the phenomenon is associated to the action of the pumping procedure of drifting electron excitation, namely,

5. A fifth one consists in a system of longitudinal acoustic phonons driven away from equilibrium by means of drifting electron excitation (presence of an electric field producing an electron current), which has been related to the creation of the so-called Saser, an acoustic laser device, with applications in computing and imaging [

6. A sixth one involving a system of LO-phonons driven away from equilibrium by means of drifting electron excitation, which displays a condensation in an off-center small region of the Brillouin zone [

Moreover, on the question of response of biological systems to MHz radiation, recently some creative and difficult experiments have been performed to probe a part of science that is poorly understood. In these experiments, microtubules—a key component of the cytoskeleton—grow from tubulin dimers through guanosine triphosphate (GTP) hydrolysis. It has been shown that, on application of 1-20 MHz radio-frequency pulses to a heat bath with tubulin dimers, microtubules can assemble orders of magnitude faster in time, suggesting that ultrafast microtubule growth occur through radio-frequency-induced resonant excitation and alignment of tubulin dimers into a cylindrical shape. Besides, the spontaneous emission of coherent 3.1-3.8 MHz signals has also been observed during the subsequent GTP-induced polymerization, and it was found that the resulting microtubules exhibit length-independent electronic and optical properties. Moreover, additional resonance levels were observed when small-molecule drugs bind to tubulin’s docking sites during radio-frequency-induced assembly. These findings can be interpreted in terms of the emergence of certain type of condensation phenomenon apparently distinct from a Fröhlich condensation [

In next section we describe the thermodynamical-statistical approach to Fröhlich condensate.

Let us consider a physical system modelling the conditions that lead to the emergence of Fröhlich effect. It is described in

An atomic model of the

The Hamiltonian consists of the energy of the free subsystems, namely, that of the free vibrations, with

For the quantum-mechanical statistical thermodynamic study of Fröhlich effect, whose results are reviewed here, we have resorted to an informational statistical thermodynamics based on the Non-equilibrium Statistical Ensemble Formalism (NESEF) [

We write for the system Hamiltonian

Finally,

Let us now consider the thermostatistics of the system characterized by the Hamiltonian of Equation (1). Nowadays, two main formalisms are available, namely, computer modelling [

Application of NESEF to a particular physical situation requires in the first place to define the set of microdynamical variables that are relevant for the treatment of the problem (the important questions of historicity and irreversibility are incorporated in the formalism from the onset [

In the most general description and for any non-equilibrium system, according to NESEF we should begin introducing all the observables of the system and their variances. However, according to the fundamental Bogoliubov’s theorem of correlation weakening and accompanying hierarchy of relaxation times [

The average of the microdynamical variables of set (5) over the non-equilibrium ensemble, which we indicate by

The relationship between the basic macrovariables of set (6) and the non-equilibrium thermodynamic variables of set (9) are what is termed as non-equilibrium equations of state, namely,

It may be noticed that the thermodynamic variable

The populations

After quite lengthy calculations it results that they follow the evolution equation given by

The six contributions on the right of this equation (rates of change of the populations generated by the different types of interactions present in the media) are:

We call Equation (15) Fröhlich contribution which is the one responsible for a super-population of the low-energy states and presence of long-lived boson coherent states in the case of items 1 to 4 above. It can be clearly noticed that the mentioned nonlinear contribution to the population of mode

The analysis presented above, after Equations (15) and (16), clearly evidence the presence in the kinetic equations for the populations of nonlinearities which are responsible for Fröhlich condensation. Using certain set of parameters, the evolution equations for the populations are solved (see [

Populations of the modes in the steady state for

An extensive analysis of the non-equilibrium statistical thermodynamics of Fröhlich condensate is reported in [

As it has been stressed in

Paul Davies [

In what refers to the processes governing consciousness in the human brain, Roger Penrose appears to have argued along a similar direction, as have been noticed in previous sections, in a kind of, say, a large-scale quantum action in brain functioning [

At this point we reproduce Penrose’s statement that, “Such a feat would be a remarkable one, almost an incredible one, for Nature to achieve by biological means. Yet I believe that the indications must be that she has done so, the main evidence coming from the fact of our own mentality. There is much to be understood about biological systems and how they achieve their magic”.

This question of large-scale quantum coherence and connection with macroscopic order has been, in some sense, partially anticipated by Herbert Fröhlich in his “The Connection between Macro- and Microphysics” [

This phenomenon has been called upon by Roger Penrose and other people as possibly having a role in consciousness, in connection with its eventual presence in microtubules in neurons.

In conclusion, as we have seen, Fröhlich condensate implies in the emergence of complex behaviour of bosons, as a result of exploring nonlinearities in the kinetic equations of evolution. There is a kind of auto-catalytic process leading to synergetic ordering, and as a consequence a decrease in informational entropy (uncertainty of information), following the fact of the consolidation of a long-range coherent macrostate. We see here the working of the microphysics—through the equations of quantum mechanics—with a subtle coupling to macrophysics through the resulting nonlinear kinetic equations, which are the average of the former over the non-equilibrium ensemble describing the expected behaviour of the whole assembly of degrees of freedom of the system.

As closing remarks, we can now recall the proclaim of the great Ludwig Boltzmann: “Thus, the general struggle for life is neither a fight for basic material ... nor for energy ... but for entropy [we say now information] becoming available by the transition from the hot sun to the cold earth” [