History: Stochastic resonance was first discovered in a study of the periodic recurrence of earth’s ice ages.

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^{}The theory developed out of an effort to understand how the earth's climate oscillates periodically between two relatively stable global temperature states, one "normal" and the other an "ice age" state. The conventional explanation was that variations in the eccentricity of earth's orbital path occurred with a period of about 100,000 years and caused the average temperature to shift dramatically. The measured variation in the eccentricity had a relatively small amplitude compared to the dramatic temperature change, however, and stochastic resonance was developed to show that the temperature change due to the weak eccentricity oscillation and added stochastic variation due to the unpredictable energy output of the sun (known as the solar constant) could cause the temperature to move in a nonlinear fashion between two stable dynamic states.

Wikipedia: Stochastic resonance is observed when noise added to a system changes the system's behaviour in some fashion. More technically, SR occurs if the signal-to-noise ratio of a nonlinear system or device increases for moderate values of noise intensity [or reversely?] It often occurs in bistable systems or in systems with a sensory threshold and when the input signal to the system is "sub-threshold". For lower noise intensities, the signal does not cause the device to cross threshold, so little signal is passed through it. For large noise intensities, the output is dominated by the noise, also leading to a low signal-to-noise ratio. For moderate intensities, the noise allows the signal to reach threshold, but the noise intensity is not so large as to swamp it. Thus, a plot of signal-to-noise ratio as a function of noise intensity shows a '∩' shape.

**Typical curve of output performance versus input noise magnitude, for systems capable of stochastic resonance.**

For small and large noise, the performance metric (e.g., SNR, mutual information, Fisher information, correlation, discrimination index) is very small, while some intermediate nonzero noise level provides optimal performance.

The word *resonance* in the term *stochastic resonance* was originally used because the signature feature of SR is that a plot of a performance measure—such as output signal-to-noise ratio (SNR)—against input noise “intensity” has a single maximum at a nonzero value. Such a plot, as shown in Figure, has a similar appearance to frequency-dependent systems that have a maximum SNR, or output response, for some *resonant frequency*. However, in the case of SR, the resonance is “noise-induced” rather than at a particular frequency.

Stochastic resonance occurs in bistable systems, when a small periodic (sinusoidal) force is applied together with a large wide band stochastic force (noise). The system response is driven by the combination of the two forces that compete/cooperate to make the system switch between the two stable states. The degree of order is related to the amount of periodic function that it shows in the system response. When the periodic force is chosen small enough in order to not make the system response switch, the presence of a non-negligible noise is required for it to happen. When the noise is small very few switches occur, mainly at random with no significant periodicity in the system response. When the noise is very strong a large number of switches occur for each period of the sinusoid and the system response does not show remarkable periodicity. Between these two conditions, there exists an optimal value of the noise that cooperatively concurs with the periodic forcing in order to make almost exactly one switch per period (a maximum in the signal-to-noise ratio).

Such a favorable condition is quantitatively determined by the matching of two time scales: the period of the sinusoid (the deterministic time scale) and the Kramers rate (i.e., the inverse of the average switch rate induced by the sole noise: the stochastic time scale). Thus the term "stochastic resonance".

- Weisstein, Eric W. "Kramers Rate." From
*MathWorld*-A Wolfram Web Resource. http://mathworld.wolfram.com/KramersRate.html The charachteristic escape rate from a stable state of a potential in the absence of signal. - Benzi, R.; Sutera, A.; and Vulpiani, A. "The Mechanism of Stochastic Resonance."
*J. Phys. A***14**, L453-L457, 1981. - Gammaitoni, L. "Stochastic Resonance E-Print Server." http://www.umbrars.com/sr/.
- Bulsara, A. R. and Gammaitoni, L. "Tuning in to Noise."
*Phys. Today***49**, 39-45, March 1996. A stochastic resonance is a phenomenon in which a nonlinear system is subjected to a periodic modulated signal so weak as to be normally undetectable, but it becomes detectable due to resonance between the weak deterministic signal and stochastic noise. The earliest definition of stochastic resonance was the maximum of the output signal strength as a function of noise.

Stochastic resonance was discovered and proposed for the first time in 1981 to explain the periodic recurrence of ice ages.^{} Since then the same principle has been applied in a wide variety of systems. Nowadays stochastic resonance is commonly invoked when noise and nonlinearity concur to determine an increase of order in the system response.

Peter Hänggi, et al.1990. Reaction-rate theory: fifty years after Kramers

The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.The classical Kramers problem (the dissociation problem): The rate of escape of a classical particle over an energy barrier is a well-posed problem as long as the potential energy features a barrier or transition-state that has to be crossed. The escape-rate problem is known to be ill-defined when the particle is trapped in a potential well which is the only point of minimum in the potential profile (which then either diverges or reaches asymptotes along the coordinated axis), see a 'brandnew', Kramers rate theory of ionization and dissociation of bound states, Alessio Zaccone, Eugene M. Terentjev,

Relativistic thermodynamics

SvaraRaderaSean A. Hayward

(Submitted on 2 Mar 1998 (v1), last revised 29 Jan 1999 (this version, v2))

A generally relativistic theory of thermodynamics is developed, based on four main physical principles: heat is a local form of energy, therefore described by a thermal energy tensor; conservation of mass, equivalent to conservation of heat, or the local first law; entropy is a local current; and non-destruction of entropy, or the local second law. A fluid is defined by the thermostatic energy tensor being isotropic. The entropy current is related to the other fields by certain equations, including a generalised Gibbs equation for the thermostatic entropy, followed by linear and quadratic terms in the dissipative (thermal minus thermostatic) energy tensor. Then the second law suggests certain equations for the dissipative energy tensor, generalising the Israel- Stewart dissipative relations, which describe heat conduction and viscosity including relativistic effects and relaxation effects. In the thermostatic case, the perfect-fluid model is recovered. In the linear approximation for entropy, the Eckart theory is recovered. In the quadratic approximation for entropy, the theory is similar to that of Israel & Stewart, but involving neither state-space differentials, nor a non-equilibrium Gibbs equation, nor non-material frames. Also, unlike conventional thermodynamics, the thermal energy density is not assumed to be purely thermostatic, though this is derived in the linear approximation. Otherwise, the theory reduces in the non-relativistic limit to the extended thermodynamics of irreversible processes due to Mueller.

The dissipative energy density seems to be a new thermodynamical field,but also exists in relativistic kinetic theory of gases.Comments: 26 pages, plain Tex, presentational changes

Subjects: General Relativity and Quantum Cosmology (gr-qc)

Cite as: arXiv:gr-qc/9803007v2

From Physics Forum threads:

SvaraRaderaAre there any relativistic thermodynamics book around or perhaps a physics review of this subject?

There is a section in Relativity, Thermodynamics and Cosmology, by Richard C. Tolman, Dover Pub, 2011 It starts on page 118 and ends on page 164.

http://books.google.fi/books/about/Relativity_Thermodynamics_and_Cosmology.html?id=1ZOgD9qlWtsC&redir_esc=y

relativistic vs non-relativistic momentum

If you wanted to be more precise you could find the ratio of classical to relativistic momentum. Or you could find the percentage difference between the two, e.g. "the relativistic momentum is 4.2% larger than the classical momentum."

Classical Physics states that:

p=mv

So, for special relativity, would momentum be defined in the same manner except m is now equal to the relativistic mass instead of the standard 'rest mass' as used in the classical equation?

...to ensure there wasnt any of the Transformations that had to apply to the velocity or anything of that nature. I was pretty sure it was just the mass change....

My comment to this is, well,look at the measurement procedure and what happen with energy then! Are they sleeping?

Quantum model for EEG, in TGD, http://tgdtheory.com/public_html/pdfpool/eegII.pdf

SvaraRaderaStochastic resonance is known to be relevant also at the neuronal level as demonstrated by the autocorrelation functions for spike sequences

exhibiting peaks at the harmonics of the signal frequency.Neuron is however far from being bistable system, and this raises the question whether bi-stability might be present at some deeper quantal level.Nerve pulses generate EEG MEs and the frequency of the nerve pulses determines the rate at which EEG MEs are generated rather than the frequency of EEG MEs. TGD inspired model of nerve pulse assigns to the resting state of cell a propagating soliton sequence and nerve pulse corresponds to a perturbation which locally transformation propagation to oscillations. The states correspond to the states of the bistable system. The system in resting state is near criticality in the sense that rotation velocity is slightly above the minimum one so that reduction of membrane potential transforms rotation motion to oscillatory motion locally. Stochastic resonances makes itself visible in the autocorrelation function of the spike sequence and in this manner also in the membrane potential of say glial cells coupling to neurons. In fact, glial cells could play the role of listener of radio turning the knob (noise level) to tune the neurons to a particular spiking frequency.

Stochastic resonance and brain at p. 19.

With motivations coming from conceptual difficulties of the proposed neuronal models, a reduction of the stochastic resonance to the quantum level, which is assumed to control the functioning of bio-systems, is developed by refi ning the quantum model for nerve pulse generation by specifying the interaction with MEs. Another key idea described in detail in [1] is that bio-systems correspond to flow equilibria for ions in the many-sheeted space-time with atomic space-time sheets having the role of a controlled system and super-conducting space-time sheets taking the role of the controlling system. The possibility that MEs generate by stochastic resonance soliton sequences associated with Josephson currents, is discussed p. 19 - 24.

http://matpitka.blogspot.com/2012/04/higgs-and-finnish-folklore.html#c7561013130073256378

SvaraRaderaI have discussed stochastic resonance in the model of EEG as a mechanism to generate "coctail party effect".

I have concentrated mainly to topics which distinguish TGD from other theories: in particular from theories which could be called generic: catastrophe theory, chaos theory, stochastic resonance, etc.. Most of the model building during last four decades relies on the assumption that gauge theories are somehow generic: you such take standard model, select gauge group, select representations for particles, select symmetry breaking and calculate beta function. This is completely mechanized waste of time but has produced impressive number of thesis works and publications. Super string theories continue the noble tradition and with some additional recipes making able to move in landscape one ends up to similar procedure. All goes wrong already at the first step: proton is predicted to decay but there is not a single thread of evidence for this.

ZEO implies two kinds of symmetries.

The zero energy states themselves become the symmetry algebra of the zero energy states. Amusing self referentiality again. This symmetry is non-local in the sense that point like particles are replaced by partonic 2-surfaces and one has something analogous to conformal algebra. It is also multilocal with respect to the partonic 2-surfaces - even at the point-like liit - as is generalization of Yangian symmetry. ZEO implies that the products for Hermitian square roots of density matrices forming a basis multiplied by powers of S-matrix S defined the algebra and this algebra generalizes the Kac-Moody type algebra.This must be extremely powerful symmetry but I am unable to deduce its consequences.

There is second symmetry. It is due to finite measurement resolution which one might call "thermodynamical". In the representation of finite measurement resolution as gauge invariance, one has complete gauge invariance so that these non-observable degrees of freedom are eliminated. In ordinary gauge invariance same holds true almost: almost because the gauge charges can be non-vanishing. For a complete gauge symmetry they would vanish. I remember that HB had talk about his idea of immensely large gauge symmetry which would break down to observed: I am not enthusiastic: proton stability kills the extension of gauge symmetry as a recipe to produce theories of everything.

Entropic stochastic resonance: the constructive role of the unevenness

SvaraRaderaP.S. Burada, G. Schmid,a, D. Reguera, J.M. Rubi, and P. H¨anggi 2008.

http://www.physik.uni-augsburg.de/theo1/hanggi/Papers/529.pdf

We demonstrate the existence of stochastic resonance (SR) in confined systems arising from entropy variations associated to the presence of irregular boundaries. When the motion of a Brownian particle is constrained to a region with uneven boundaries, the presence of a periodic input may give rise to a peak in the spectral amplification factor and therefore to the appearance of the SR phenomenon. We have proved that the amplification factor depends on the shape of the region through which the particle moves and that by adjusting its characteristic geometric parameters one may optimize the response of the system. The situation in which the appearance of such entropic stochastic resonance (ESR) occurs is common for small-scale systems in which confinement and noise play an prominent role. The novel mechanism found could thus constitute an important tool for the characterization of these systems and can put to use for controlling their basic properties.

Up to now, the phenomenon of SR has been observed mainly in systems dominated by the presence of a purely energetic potential or possessing some dynamical threshold [1]. However, when scaling down the size of a system, the free energy rather than the internal energy becomes the most appropriate potential, and there are cases in which changes in the free energy are mainly due to entropy variations [11,13–18]. This is what occurs in constrained systems. In the case of a Brownian particle moving in a confined medium, entropy variations contribute to changes in the free energy and may under some circumstances become its leading contribution [12,13,17,18].

Geometric Stochastic Resonance

SvaraRaderaPulak Kumar Ghosh,et-al, 2010 http://dml.riken.jp/pub/nori/pdf/PRL_104_020601_2010.pdf

A Brownian particle moving across a porous membrane subject to an oscillating force exhibits stochastic resonance with properties which strongly depend on the geometry of the confining cavities on the two sides of the membrane. Such a manifestation of stochastic resonance requires neither energetic nor entropic barriers, and can thus be regarded as a purely geometric effect. The magnitude of this effect is sensitive to the geometry of both the cavities and the pores, thus leading to distinctive optimal synchronization conditions.

... particles are often confined to constrained geometries, such as interstices, pores, or channels, whose size and shape can affect the SR mechanism [4]. Indeed, smooth confining geometries can be modeled as entropic (i.e., noise or temperature dependent) potentials [5], capable of influencing the response of the system to an external driving force [6].