Stochastic resonance

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Stochastic_resonance 

Stochastic resonance (SR) is a mathematical mechanism and behavior of nonlinear systems (that is, systems in which the change of the output is not proportional to the change of the input) where random (stochastic) fluctuations in the microstate of a system (that is, its specific configuration, including the precise positions and momenta of all its individual particles or components) cause deterministic (that is, non-random) changes in a macrostate (that is, a subset of the system's microstates).

This occurs when the nonlinear nature of the system amplifies certain (resonant) portions of the fluctuations, while not amplifying other portions of the noise. The nonlinear system, immersed in a certain level of stochastic background noise, becomes sensitive to external perturbations that would be too weak to influence it in the absence of such noise.

Originally proposed in the context of climate dynamics, over time it has become important in numerous fields that study a wide variety of systems, particularly in information theory and in neuroscience. Phenomena attributable to stochastic resonance have also been observed in other types of physical systems, such as chemical reactions, quantum systems, and industrial processes. Stochastic resonance is also closely related to the concept of dithering in signal analysis, although how similar or how different the two concepts are depends on the particular definition considered.

Stochastic resonance was discovered and proposed for the first time in 1981 to explain the periodic recurrence of ice ages. Nowadays stochastic resonance is commonly invoked when noise and nonlinearity concur to determine an increase of order in a system's response.

See the Review of Modern Physics article "Stochastic resonance" for a comprehensive overview of stochastic resonance.

History

The mechanism of stochastic resonance was first described in the early 1980s by the Italian physicists Roberto Benzi, Alfonso Sutera, and Angelo Vulpiani, who (with the additional participation of Giorgio Parisi) immediately applied it to climatology, in order to explain how small variations in the motion of the Earth (the so-called Milankovitch cycles) can cause large variations in the Earth's climate (in particular, the transition from glacial periods to interglacial periods, and vice versa). According to Parisi's account, the name "stochastic resonance" was coined by Benzi during a conference. At the same time, a very similar explanation was also proposed by the Belgian physicist Catherine Nicolis.

An initial experimental verification was found as early as 1983 in a bistable electronic circuit, and in 1988 in a laser system. In the early 1990s, the first works appeared in which it was hypothesized that stochastic resonance played an important role in neuronal dynamics, a concept now confirmed.

Climatological interpretation

In the original works of Benzi, Parisi, Sutera, and Vulpiani, the potential depending on the Earth's mean temperature (i.e., the variable corresponding to ${\displaystyle x}$) was linked to the albedo of the Earth, that is, the fraction of incoming solar radiation that is reflected back into space rather than absorbed by the planet. It depends on numerous factors closely related to the Earth's climate, the main ones being the extent of the ice sheets and cloud cover. In general it was assumed that the albedo tends to a maximum both for extremely low temperatures (since the planet would be completely covered by highly reflective ice) and for high temperatures (since high temperature is associated with high evaporation, and therefore with extensive cloud cover, which is also reflective), while the two stable states of minimum albedo were associated with glacial periods and interglacial periods.

Information theory

In information theory, SR can be used to reveal weak signals. When a signal that is normally too weak to be detected by a sensor can be boosted by adding white noise to the signal, which contains a wide spectrum of frequencies. The frequencies in the white noise corresponding to the original signal's frequencies will resonate with each other, amplifying the original signal while not amplifying the rest of the white noise – thereby increasing the signal-to-noise ratio, which makes the original signal more prominent. Further, the added white noise can be enough to be detectable by the sensor, which can then filter it out to effectively detect the original, previously undetectable signal.

This phenomenon of boosting undetectable signals by resonating with added white noise extends to many other systems – whether electromagnetic, physical or biological – and is an active area of research.

Technical description

Stochastic resonance (SR) 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. 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 contains a peak.

Strictly speaking, 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 timescales: the period of the sinusoid (the deterministic time scale) and the Kramers rate (i.e., the average switch rate induced by the sole noise: the inverse of the stochastic time scale.

Simplified model

Generic shape of the potential ${\displaystyle U(x)}$

Below is a didactic toy model capable of capturing the essential aspects of the stochastic resonance mechanism.

Starting equation

Consider a time-dependent physical variable ${\displaystyle x(t)}$ (which, in the original climate papers, was the mean temperature of the Earth), described by a stochastic differential equation of the Langevin equation type:

$${\displaystyle \frac{dx}{dt}=-\frac{\mathrm{\partial }U}{\mathrm{\partial }x}+f(t)+ϵ\cos (\omega t),}$$

where ${\displaystyle U(x)}$ is a potential function determining the system dynamics, ${\displaystyle f(t)}$ is a stochastic forcing corresponding to background noise, and ${\displaystyle ϵ\cos (\omega t)}$ is the external periodic perturbation, of small amplitude (${\displaystyle ϵ\ll 1}$), which in the climate studies corresponded to the Milankovitch cycles. It is assumed that ${\displaystyle U(x)}$ has a double-well shape (describable by a fourth-degree polynomial), i.e., it possesses two stable minima ${\displaystyle x_{-}}$ and ${\displaystyle x_{+}}$ (glacial and interglacial periods) separated by an unstable maximum ${\displaystyle x_0}$, corresponding to a potential barrier ${\displaystyle \mathrm{\Delta }U}$ (for simplicity it is assumed that the two minima lie at the same level ${\displaystyle U(x_{\pm })}$). For ${\displaystyle f(t)}$ the simplest possible form is considered, namely that of Gaussian noise, with zero mean and Dirac delta correlation: ${\displaystyle ⟨f(t)⟩=0}$ and ${\displaystyle ⟨f(t_1)f(t_2)⟩=2D\delta (t_2-t_1)}$.

In the absence of periodic perturbations (${\displaystyle ϵ=0}$), it is well known from the theory of Markov processes that the variable ${\displaystyle x(t)}$ will fluctuate around the minimum points, with a variance proportional to the Gaussian noise intensity ${\displaystyle D}$. Occasionally the fluctuations will be strong enough to allow crossing of the potential barrier, and therefore the transition from one minimum state to the other. In the limit ${\displaystyle D\ll \mathrm{\Delta }U}$, the mean transition frequency is given by the so-called Kramers formula:

$${\displaystyle r_{\pm }=\frac{1}{2\pi }\sqrt{-U^{{}^{″}}(x_0)U^{{}^{″}}(x_{\pm })}{e}^{-\frac{\mathrm{\Delta }U}{D}},}$$ where ${\displaystyle U^{{}^{″}}(x)}$ is the second derivative of ${\displaystyle U(x)}$.

Stochastic resonance

In the presence of periodic perturbations, the potential ${\displaystyle U}$ is replaced by the generalized potential ${\displaystyle W(x,t)=U(x)-ϵ\cos \omega t}$. This means that the two potential wells vary in height, rising and falling, and therefore the barrier that the system must overcome to jump from one minimum to the other will at some moments be higher and at others lower. However, the oscillations are of small amplitude, so they are not able to completely remove the barrier and thus allow the system to transition between states on their own: the idea of stochastic resonance is that, if the perturbation frequency ${\displaystyle \omega }$ is in some way comparable with the mean transition frequency ${\displaystyle r_{\pm }}$, then the random fluctuations of the system will tend to synchronize with the external oscillations, making the transition more likely.

Using the explicit form of the potential ${\displaystyle U(x)=-\frac{1}{2}{x}^2+\frac{1}{4}{x}^4}$, it can be shown that, at least to first order approximation, the mean value of ${\displaystyle x(t)}$ becomes a periodic function of time with the same frequency as the external forcing:

$${\displaystyle ⟨x(t)⟩=A\cos (\omega t+\varphi ),}$$

where the amplitude ${\displaystyle A}$ and the phase ${\displaystyle \varphi }$ are given by the following relations:

Typical behavior of the amplitude ${\displaystyle A}$ as a function of the noise intensity ${\displaystyle D}$.

$${\displaystyle A=\frac{ϵ⟨x^2{⟩}_0}{D}\frac{2r_{\pm }}{\sqrt{r_{\pm }^2+{\omega }^2/4}},\varphi =-\arctan \left(\frac{\omega }{2r_{\pm }}\right);}$$

with ${\displaystyle ⟨x^2{⟩}_0}$ the variance of ${\displaystyle x}$ in the absence of external perturbations (dependent on the noise intensity ${\displaystyle D}$).

The fundamental aspects are therefore:

• the system oscillations tend, on average, to synchronize with the external perturbations;
• the amplification effect is negligible unless the perturbation frequency ${\displaystyle \omega }$ is comparable with the transition frequency ${\displaystyle r_{\pm }}$ (dependent on the noise intensity), and it is maximal for ${\displaystyle \omega =2r_{\pm }}$;
• for fixed perturbation amplitude ${\displaystyle ϵ}$ and frequency ${\displaystyle \omega }$, the amplitude ${\displaystyle A}$ is, for low noise intensities, a sharply increasing function of ${\displaystyle D}$, reaching a maximum peak at an intermediate noise level, and then becoming a decreasing function of ${\displaystyle D}$ for large noise values.

Suprathreshold

Suprathreshold stochastic resonance is a particular form of stochastic resonance, in which random fluctuations, or noise, provide a signal processing benefit in a nonlinear system. Unlike most of the nonlinear systems in which stochastic resonance occurs, suprathreshold stochastic resonance occurs when the strength of the fluctuations is small relative to that of an input signal, or even small for random noise. It is not restricted to a subthreshold signal, hence the qualifier.

Neuroscience, psychology and biology

Main article: Stochastic resonance (sensory neurobiology)

Stochastic resonance has been observed in the neural tissue of the sensory systems of several organisms. Computationally, neurons exhibit SR because of non-linearities in their processing. SR has yet to be fully explained in biological systems, but neural synchrony in the brain (specifically in the gamma wave frequency) has been suggested as a possible neural mechanism for SR by researchers who have investigated the perception of "subconscious" visual sensation. Single neurons in vitro including cerebellar Purkinje cells and squid giant axon could also demonstrate the inverse stochastic resonance, when spiking is inhibited by synaptic noise of a particular variance.

Medicine

SR-based techniques have been used to create a novel class of medical devices for enhancing sensory and motor functions such as vibrating insoles especially for the elderly, or patients with diabetic neuropathy or stroke.

Stochastic resonance has found noteworthy application in the field of image processing.

Signal analysis

A related phenomenon is dithering, applied to analog signals before analog-to-digital conversion. Stochastic resonance can be used to measure transmittance amplitudes below an instrument's detection limit. If Gaussian noise is added to a subthreshold (i.e., immeasurable) signal, then it can be brought into a detectable region. After detection, the noise is removed. A fourfold improvement in the detection limit can be obtained.

Glueball

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Glueball 

In particle physics, a glueball (also gluonium, gluon-ball) is a hypothetical composite particle. It consists solely of gluons, without valence quarks. Such a state is possible because gluons carry color charge and experience the strong interaction between themselves. Glueballs are extremely difficult to identify in particle accelerators, because they mix with ordinary meson states. In pure gauge theory, glueballs are the only states of the spectrum and some of them are stable.

Theoretical calculations show that glueballs should exist at energy ranges accessible with current collider technology. However, due to the aforementioned difficulty (among others), they have so far not been observed and identified with certainty, although phenomenological calculations have suggested that an experimentally identified glueball candidate, denoted f₀(1710), has properties consistent with those expected of a Standard Model glueball.

The prediction that glueballs exist is an essential prediction of QCD as part of the Standard Model of particle physics that has not yet been unambiguously confirmed experimentally.

Experimental evidence was announced in 2021, by the TOTEM collaboration at the LHC in collaboration with the DØ collaboration at the former Tevatron collider at Fermilab, of odderon (a composite gluonic particle with odd C-parity) exchange. This exchange, associated with a quarkless three-gluon vector glueball, was identified in the comparison of proton–proton and proton–antiproton scattering. In 2024, the X(2370) particle was determined to have mass and spin parity consistent with that of a glueball. However, other exotic particle candidates such as a tetraquark could not be ruled out.

Properties

In principle, it is theoretically possible for all properties of glueballs to be calculated exactly and derived directly from the equations and fundamental physical constants of quantum chromodynamics (QCD) without further experimental input. So, the predicted properties of these hypothetical particles can be described in exquisite detail using only Standard Model physics that have wide acceptance in the theoretical physics literature. But, there is considerable uncertainty in the measurement of some of the relevant key physical constants, and the QCD calculations are so difficult that solutions to these equations are almost always numerical approximations (calculated using several very different methods). This can lead to variation in theoretical predictions of glueball properties, like mass and branching ratios in glueball decays.

Constituent particles and color charge

Theoretical studies of glueballs have focused on glueballs consisting of either two gluons or three gluons, by analogy to mesons and baryons that have two and three quarks respectively. As in the case of mesons and baryons, glueballs would be QCD color charge neutral. The baryon number of a glueball is zero.

Total angular momentum

Double-gluon glueballs can have total angular momentum J = 0 (which are either scalar or pseudo-scalar) or J = 2. Triple-gluon glueballs can have total angular momentum J = 1 (vector boson) or J = 3. All glueballs have integer total angular momentum that implies that they are bosons rather than fermions.

Glueballs are the only particles predicted by the Standard Model with total angular momentum (J) (sometimes called intrinsic spin) that could be either 2 or 3 in their ground states, although mesons made of two quarks with J = 0 and J = 1 with similar masses have been observed and excited states of other mesons can have these values of total angular momentum.

Electric charge

All glueballs would have an electric charge of zero, as gluons themselves do not have an electric charge.

Mass and parity

Glueballs are predicted by quantum chromodynamics to be massive, despite the fact that gluons themselves have zero rest mass in the Standard Model. Glueballs with all four possible combinations of quantum numbers P (spatial parity) and C (charge parity) for every possible total angular momentum have been considered, producing at least fifteen possible glueball states including excited glueball states that share the same quantum numbers but have differing masses with the lightest states having masses as low as 1.4 GeV/c² (for a glueball with quantum numbers J = 0, P = +1, C = +1, or equivalently J^PC = 0⁺⁺), and the heaviest states having masses as great as almost 5 GeV/c² (for a glueball with quantum numbers J = 0, P = +1, C = −1, or J^PC = 0⁺⁻).

These masses are on the same order of magnitude as the masses of many experimentally observed mesons and baryons, as well as to the masses of the tau lepton, charm quark, bottom quark, some hydrogen isotopes, and some helium isotopes.

Stability and decay channels

Just as all Standard Model mesons and baryons, except the proton, are unstable in isolation, all glueballs are predicted by the Standard Model to be unstable in isolation, with various QCD calculations predicting the total decay width (which is functionally related to half-life) for various glueball states. QCD calculations also make predictions regarding the expected decay patterns of glueballs. For example, glueballs would not have radiative or two photon decays, but would have decays into pairs of pions, pairs of kaons, or pairs of eta mesons.

Practical impact on macroscopic low energy physics

Feynman diagram of a glueball (G) decaying to two pions (π). Such decays help the study of and search for glueballs.

Standard Model glueballs are extremely ephemeral (decaying almost immediately into more stable decay products) and are only generated in high energy physics. Thus in the natural conditions found on Earth that humans can easily observe, glueballs arise only synthetically. They are scientifically notable mostly because they are a testable prediction of the Standard Model, and not because of phenomenological impact on macroscopic processes, or their engineering applications.

Lattice QCD simulations

Lattice QCD provides a way to study the glueball spectrum theoretically and from first principles. Some of the first quantities calculated using lattice QCD methods (in 1980) were glueball mass estimates. Morningstar and Peardon computed the masses of the lightest glueballs in QCD without dynamical quarks in 1999. The three lowest states are tabulated below. The presence of dynamical quarks would slightly alter these data, but also makes the computations more difficult. Since that time calculations within QCD (lattice and sum rules) find the lightest glueball to be a scalar with mass in the range of about 1000–1700 MeV/c². Lattice predictions for scalar and pseudoscalar glueballs, including their excitations, were confirmed by Dyson–Schwinger/Bethe–Salpeter equations in Yang–Mills theory.

JPC	mass
0++	1730±80 MeV/c2
2++	2400±120 MeV/c2
0−+	2590±130 MeV/c2

Experimental candidates

Particle accelerator experiments are often able to identify unstable composite particles and assign masses to those particles to a precision of approximately 10 MeV/c², without being able to immediately assign to the particle resonance that is observed all of the properties of that particle. Scores of such particles have been detected, although particles detected in some experiments but not others can be viewed as doubtful.

Many of these candidates have been the subject of active investigation for at least eighteen years. The GlueX experiment has been specifically designed to produce more definitive experimental evidence of glueballs.

Some of the candidate particle resonances that could be glueballs, although the evidence is not definitive, include the following.

Glueball candidates

• X(3020) observed by the BaBar collaboration is a candidate for an excited state of the J^PC = 2⁻⁺, 1⁺⁻, or 1⁻⁻ glueball states with a mass of about 3.02 GeV/c².

Scalar glueball candidates

Various candidates for scalar glueballs were identified by Ochs:

• f₀(500) also known as σ – the properties of this particle are possibly consistent with a glueball of mass 1000 MeV/c² or 1500 MeV/c².
• f₀(980) – the structure of this composite particle is consistent with the existence of a light glueball.
• f₀(1370) – existence of this resonance is disputed but is a candidate for a glueball–meson mixing state.
• f₀(1500), f₀(1710) – existence of these resonances is undisputed but their statuses as glueball–meson mixing states or pure glueballs is not well established.

Other candidates

• Gluon jets at the LEP experiment show a 40% excess over theoretical expectations of electromagnetically neutral clusters, which suggests that electromagnetically neutral particles expected in gluon-rich environments such as glueballs are likely to be present.