Much like these rocks! More recently the concept of the Lyapunov exponent (related to Lyapunov's First Method of discussing stability) has received wide interest in connection with chaos theory. ), Correspondingly, a time-discrete linear state space model. x Notably, this is the case for anatase, a metastable polymorph of titanium dioxide, which despite commonly being the first phase to form in many synthesis processes due to its lower surface energy, is always metastable, with rutile being the most stable phase at all temperatures and pressures. a different set of quantum interactions that could occur. [3][4][5][6][7] ) , | t M {\displaystyle w'\left(\eta \right)\,} Instead, dependent on specifics like how far your field is from the true equilibrium state and how high the barrier is preventing you from leaving the false minimum that youre stuck in, theres a certain probability that you can spontaneously leave your unstable equilibrium state and find yourself, all of a sudden, in a more stable (or even the true) minimum of your quantum system. u 0 ( Nevertheless, you can still leave the unstable equilibrium state and arrive at the ground state via quantum tunneling: the source of the famed. {\displaystyle e} = in a chemical reaction or phase transition.The chemical potential of a species in a mixture is defined as the rate of change of free energy of a thermodynamic system with respect to the change in the number of Drag and drop terms to their correct position in the atmospheric stability diagram below: What does the Lifting Condensation Level (LCL) represent? Math. {\displaystyle c\eta =\Psi \,} Types of equilibriumWe can divide the equilibrium into two categories with respect to the state of a body1. gives. {\displaystyle x_{e}} This is what we call the lowest-energy state, or the ground-state, of a system. Copy a link to the article entitled http://Is%20the%20Universe%20fundamentally%20unstable? is required; otherwise for example But we can know that not only would the ones we see today cease to exist, but that wherever this transition occurred, it would propagate outward at the speed of light, infecting space as it expanded with a great bubble of destruction. Property of a dynamical system where solutions near an equilibrium point remain so, This article is about asymptotic stability of nonlinear systems. Carbon makes up only about 0.025 percent of Earth's crust. Dynamic equilibriumwhen a body is in a state of uniform motion and the resultant total force and total torque on the body are zero then it is said to be in dynamic equilibrium. It will help in every subject. So for slight displacement, There is no force, potential energy diagram for the Neutral Equilibrium, We can see lots of these examples in gravitational PE, Spring PE, and electrical PE, How to solve the Potential energy function equilibrium Problems, (1) Potential energy function must be given for the problem(2) Differentiate with respect to the variable(3) To find the equilibrium points, put dU/dx=0 and solve for the values of x(4) Perform second differentiation of the Potential energy function(5) Find the value of the second derivative for the equilibrium points(6) If it is positive, we are having stable(If it is zero, it is neutral equilibrium(if it is negative ,it is unstable equilibrium, (1) The potential energy function of the object confined to move along x-axis x> 0 is given below$U(x)=U_{0}[(\frac{k}{x})^{12}-2(\frac{k}{x})^{6}]$Here k > 0 and $U_{0}$ > 0. [2][3][4] Examples include the behavior of water suspended above oil in the gravity of Earth,[3] mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions,[5] supernova explosions in which expanding core gas is accelerated into denser shell gas,[6][7] instabilities in plasma fusion reactors and[8] inertial confinement fusion. 2 1 A {\displaystyle \Re \left\{\cdot \right\}\,} Travel the Universe with astrophysicist Ethan Siegel. [3] The important insight by G. I. Taylor was his realisation that this situation is equivalent to the situation when the fluids are accelerated, with the less dense fluid accelerating into the denser fluid. {\displaystyle x(t)=\phi (t)} t {\displaystyle \nabla \times {\textbf {u}}'=0\,} c Three isotopes occur naturally, 12 C and 13 C being stable, while 14 C is a Take a look at this crazy potential energy curve. t x During a metastable state of finite lifetime, all state-describing parameters reach and hold stationary values. The higher it is the more likely the object is to topple over if it is pushed. An electronic-only journal, JCAP is jointly owned by IOP Publishing and SISSA. {\displaystyle g} x {\displaystyle {\mathcal {D}}} | , (Plotted against distance), The Potential energy curve, interestinglyis a lot like a ball in a bowl , (Forgive my poor skills with photoshop ;P). They are G There are many things we can learn about the atmosphere from Skew-T Log-P diagrams. Of course, if the temperature is different at different heights, the air is unstable thermodynamically. e If you draw out any potential, it will have a profile where at least one point corresponds to the lowest-energy, or true vacuum, state. , For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid; for A close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes. t Equilibrium on a Potential Energy vs. . 1 converge to ) December 01, 2022 December '22. V The same concept of Potential Energy is equally useful in Physics and Chemistry.At x(0) and x(2) a body (or chemical) would be safe. $U(x)=U_{0}[(\frac{k}{x})^{12}-2(\frac{k}{x})^{6}]$, $\frac{dU}{dx}=U_{0}[-12\frac{k^{12}}{x^{13}}+12\frac{k_{6}}{x^{7}}]$Now for equilibrium position,$\frac{dU}{dx}=0$$U_{0}[-12\frac{k^{12}}{x^{13}}+12\frac{k_{6}}{x^{7}}]=0$or x=k, $\frac{d^{2}U}{dx^{2}}=U_{0}[+156\frac{k^{12}}{x^{14}}-84\frac{k_{6}}{x^{8}}]$, $\frac{d^{2}U}{dx^{2}}=72\frac{U_{0}}{k^{2}}$, Which is clearly positive as k > 0 and $U_{0}$ > 0So it is the case of Stable equilibrium, (2) The potential energy U of an object of unit mass in a one-dimensional conservative force field is given by. The thermodynamic free energy is a concept useful in the thermodynamics of chemical or thermal processes in engineering and science. Both of these types of decays change the elements atomic number, yielding a new element different from the original, and result in a lower mass for the products than for the reactants. = x An intensive property is a physical quantity whose value does not depend on the amount of substance which was measured. {\displaystyle \nabla ^{2}\psi =0.\,} Ask Ethan: Would a false vacuum state of the Universe lead to our destruction? Lyapunov's realization was that stability can be proven without requiring knowledge of the true physical energy, provided a Lyapunov function can be found to satisfy the above constraints. Well, you know what the ultimate quantum system is? If it is, then great: the Universe will be stable forever and ever, as theres no lower-energy state for it to ever quantum tunnel into. = The size of our visible Universe (yellow), along with the amount we can reach (magenta) if we left, today, on a journey at the speed of light. Conceptually, the meanings of the above terms are the following: The trajectory (only its derivatives) because A common type of metastability is isomerism. {\displaystyle \gamma \,} Reaction intermediates are relatively short-lived, and are usually thermodynamically unstable rather than metastable. z = ( Learn how your comment data is processed. = z Microsoft pleaded for its deal on the day of the Phase 2 decision last month, but now the gloves are well and truly off. e Next, because of the translational invariance of the system in the x-direction, it is possible to make the ansatz, where To understand the implications of this result in full, it is helpful to consider the case of zero surface tension. Moreover, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays irrotational, hence With this knowledge, the location of the tropopause, given by its pressure level, can be determined by examining a plotted sounding. Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (a=0), but rather, are in static equilibrium with their environment. {\displaystyle \alpha \,} Account Manager, Stable Value . Recall that the standard temperature decreases with height within the troposphere, but becomes isothermal with height within the the tropopause, and increases with height in the stratosphere. D But this is wrong, since Molecular vibrations and thermal motion make chemical species at the energetic equivalent of the top of a round hill very short-lived. This says that Of course, since transitions from a metastable state are not impossible (merely less likely), the electron will eventually decay to a less energetic state, typically by an electric quadrupole transition, or often by non-radiative de-excitation (e.g., collisional de-excitation). [16], The inviscid two-dimensional RayleighTaylor (RT) instability provides an excellent springboard into the mathematical study of stability because of the simple nature of the base state. This page was last edited on 3 December 2022, at 21:55. Carbon (from Latin carbo 'coal') is a chemical element with the symbol C and atomic number 6. Lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems. z for all trajectories Now define Sometimes, there are openings in those lower-energy states, and those higher-energy electrons will spontaneously cascade downward, emitting energy in the process. The definition below provides this, using an alternate language commonly used in more mathematical texts. G x / z , t Here weve provided just the basics to get you started. x Monthly (2016) 128, no. is Lyapunov stable. There is no lower-energy state, but there are semi-transient signals in the brain that persist for a while and are different than the usual equilibrium state. The inviscid two-dimensional RayleighTaylor (RT) instability provides an excellent springboard into the mathematical study of stability because of the simple nature of the base state. V e {\displaystyle z=\eta (x,t),\,} Quantum mechanical properties of the We will guide you on how to place your essay help, proofreading and editing your draft fixing the grammar, spelling, or formatting of your paper easily and cheaply. The stability or metastability of a given chemical system depends on its environment, particularly temperature and pressure. Procedure for CBSE Compartment Exams 2022, Find out to know how your mom can be instrumental in your score improvement, (First In India): , , , , Remote Teaching Strategies on Optimizing Learners Experience, MP Board Class 10 Result Declared @mpresults.nic.in, Area of Right Angled Triangle: Definition, Formula, Examples, Composite Numbers: Definition, List 1 to 100, Examples, Types & More, Potential Energy is often ignored. {\displaystyle w} < {\displaystyle c\eta =\Psi \,} Lyapunov, in his original 1892 work, proposed two methods for demonstrating stability. 4 (1959) 267270, p. 269. Theory of Stability of Motion, Moscow 1952 (Gostekhizdat) Chap II para 4 (Russian) Engl. {\displaystyle x(t)=\phi (t)} The analysis in the previous section breaks down when the amplitude of the perturbation is large. If that happens anywhere, every bound structure, from protons on up, will be destroyed in a "bubble of destruction" propagating outward at the speed of light. ) t Heres why we dont want to live in the former. In dry conditions, snow slopes act similarly to sandpiles. x 0 If the force is restoring i,e direction is towards the equilibrium position then it is a stable equilibrium. [8], Consider an autonomous nonlinear dynamical system, where This turns out to be a property of all stable mechanical systems. This example is from Chemistry. Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. Unlike in the purely classical case, this can happen spontaneously, with no outside, energetic influence or impetus required. i Pures Appl. (c) If the speed of the object at equilibrium position is 4m/s,what is the amplitude of the oscillations, So -(2x-4) =0 or x=2 m is the equilibrium position, Let the object is displaced with x0 from equilibrium position,then force, So Force is directly proportional to displacement and it is restoring one also. 0 Biomass is plant-based material used as fuel to produce heat or electricity.Examples are wood and wood residues, energy crops, agricultural residues, and waste from industry, farms and households. 8. {\displaystyle x=0} The domain of the problem is the following: the fluid with label 'L' lives in the region The most obvious intensive quantities are ratios of extensive quantities. In the non-linear phase, perturbation amplitude is too large for a linear approximation, and non-linear equations are required to describe fluid motions. Here, we will look at the dimensional formula of pressure and how to find the, Suppose two charges q1 and q2 are moving with velocities v1 and v2 $respectively. This bubble of destruction would be unnoticeable until it arrived. For several decades the theory of stability sank into complete oblivion. t and Ask Ethan: Where does quantum uncertainty come from? = {\displaystyle M=M^{\textsf {T}}} Aggregated systems of subatomic particles described by quantum mechanics (quarks inside nucleons, nucleons inside atomic nuclei, electrons inside atoms, molecules, or atomic clusters) are found to have many distinguishable states. x So equilibrium would be unstable. {\displaystyle \varepsilon } The abundance of states is more prevalent as the systems grow larger and/or if the forces of their mutual interaction are spatially less uniform or more diverse. x p A digital circuit is supposed to be found in a small number of stable digital states within a certain amount of time after an input change. V x Rather, it can roll into a valley thats still lower than where it started, but that doesnt represent the true ground state of the system. {\displaystyle z=\eta } If youre a classical system, the only solution is Sisyphean: you have to input enough energy into your system irrespective of whether thats kinetic energy, chemical energy, electrical energy, etc. In fact, the potential energy curve for this situation is very similar to that of the spring and block example above! ), one can formulate similar definitions of stability near an arbitrary solution . This is a common situation for the droplets of atmospheric clouds. e ( The particular motion or kinetics of the atoms involved has resulted in getting stuck, despite there being preferable (lower-energy) alternatives. = ) 1 whose stability is equivalent to the stability of the original solution = c But if it's unstable, then the quantum vacuum may decay to a more stable. But it is not left to itself; the sun is always shining (during the day). Higher energy isomers are long lived because they are prevented from rearranging to their preferred ground state by (possibly large) barriers in the potential energy. is the Atwood number. Much like these rocks! (The relevant Lyapunov function is {\displaystyle V(t)\leq V(0)} D {\displaystyle 0\leq z<\infty \,} If the ball is only slightly pushed, it will settle back into its hollow, but a stronger push may start the ball rolling down the slope. Math. 1 ) . p [5] Of all these other states, the metastable states are the ones having lifetimes lasting at least 102 to 103 times longer than the shortest lived states of the set.[6]. This in turn creates additional vorticity, leading to further misalignment. Later, it was borrowed for the study of decision-making and information transmission systems. Lee, S.N. } such that. < [17] This is the equilibrium state that exists before any perturbation is added to the system, and is described by the mean velocity field Thus, If the ball is only slightly pushed, it will settle back into its hollow, but a stronger push may start the ball rolling down the slope. In the stable configuration, the vorticity, and thus the induced velocity field, will be in a direction that decreases the misalignment and therefore stabilizes the system.[16][19]. The vacuum state of the Universe, remember, depends on the fundamental laws, quanta, and constants that underlie our Universe. is locally stable.) g ) with the boundary conditions In chemistry and physics, metastability denotes an intermediate energetic state within a dynamical system other than the system's state of least energy.A ball resting in a hollow on a slope is a simple example of metastability. z ). {\displaystyle e} Static equilibriumIf the total force and total torque on the body are zero and the body is in the state of rest then its equilibrium is termed as static equilibrium.For example All stationary bodies. Magnetic force is used to counteract the effects of the gravitational force and any other forces.. I. Turbulent Upflow Plumes", "Numerical Simulations of the Magnetic RayleighTaylor Instability in the Kippenhahn-Schlter Prominence Model. But its nothing the Universe hasnt already seen, and survived. 1 , . x together, Substituting the second interfacial condition document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); document.getElementById( "ak_js_2" ).setAttribute( "value", ( new Date() ).getTime() ); This site uses Akismet to reduce spam. ( G [1] A. M. Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium. And balls placed on a hilly surface will tend to roll down into the valley below, coming to rest at the bottom: at the lowest possible elevation that their initial conditions enabled them to reach. A metastable state is then long-lived (locally stable with respect to configurations of 'neighbouring' energies) but not eternal (as the global minimum is). Left to itself infinitely long, the air would all come to the same temperature. Embiums Your Kryptonite weapon against super exams! This is non-autonomous because the input An introduction to nuclear waste immobilisation. w 2 [3] This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion. In meteorology, convective available potential energy (commonly abbreviated as CAPE), is the integrated amount of work that the upward (positive) buoyancy force would perform on a given mass of air (called an air parcel) if it rose vertically through the entire atmosphere.Positive CAPE will cause the air parcel to rise, while negative CAPE will cause the air parcel to sink. . V 1999, pp. ) The change in the free energy is the maximum amount of work that a thermodynamic system can perform in a process at constant temperature, and its sign indicates whether the process is thermodynamically favorable or forbidden. If the system loses energy over time and the energy is never restored then eventually the system must grind to a stop and reach some final resting state. When in the unstable configuration, for a particular harmonic component of the initial perturbation, the torque on the interface creates vorticity that will tend to increase the misalignment of the gradient vectors. x a small displacement of the ball toward either side will result in a force returning the ball to the original position. A {\displaystyle {\mathcal {A}}\,} 0 {\displaystyle \{A_{1},\dots ,A_{m}\}} Even in empty space, this vacuum energy is non-zero, and what appears to be the ground state in one region of curved space will look different from the perspective of an observer where the spatial curvature differs. ) The growth then becomes non-linear as the spikes and bubbles of the instability tangle and roll up into vortices. Dynamic equilibrium. Intensive properties. In a homogeneous system divided into two halves, all its extensive properties, in particular its volume and its mass, are divided into two halves. It is possible for an entire large sand pile to reach a point where it is stable, but the addition of a single grain causes large parts of it to collapse. Share Is the Universe fundamentally unstable? The electron's mass is approximately 1/1836 that of the proton. m , asset allocation, all asset. x if the force is away i,e direction is away from the equilibrium position then it is an unstable equilibrium. t {\displaystyle B=A/c.\,}. {\displaystyle x(t)=x_{e}} {\displaystyle \left(D^{2}-\alpha ^{2}\right)\Psi _{i}=0,\,} This latter option, unfortunately, is not so great. It can be converted to graphite (plus leftover kinetic energy), but only after overcoming an activation energy an intervening hill. on ( {\displaystyle z=-\infty \,} . Students think that as long as you know Newtons equations of motion, you dont need to know anything else about the problem!However, a simple differential of the Potential Energy can make life easier even in complicated situations.Thats because , The classic case of a block attached to a spring . in the upper region, and denotes the real part of the complex valued expression between brackets. at However, finding a function that gives the precise energy of a physical system can be difficult, and for abstract mathematical systems, economic systems or biological systems, the concept of energy may not be applicable. / 2 c Smashing things together at unprecedented energies sounds dangerous. So for slight displacement, the restoring force should act on the body, potential energy diagram for the StableEquilibrium, For unstable equilibrium, the second derivative should be Negative. There are certain modes or positions of special interest are called equilibrium. = are bounded. This is shown by-If the potential energy increases along x, the Force is negative, pulling it away from that direction! Note that the RT instability is not to be confused with the PlateauRayleigh instability (also known as Rayleigh instability) of a liquid jet. Being excited of an energy above the ground state it will eventually decay to a more stable state, releasing energy. The interest in it suddenly skyrocketed during the Cold War period when the so-called "Second Method of Lyapunov" (see below) was found to be applicable to the stability of aerospace guidance systems which typically contain strong nonlinearities not treatable by other methods. Metastable polymorphs of silica are commonly observed. = G . Third edition, Elsevier, Amsterdam, p.323 (2019), Process Chemistry in the Pharmaceutical Industry. Equilibrium is that point on the potential curve where Force is zero, i.e.-. In regions where a particular state is not the most stable, it may still be metastable. Being "stuck" in a thermodynamic trough without being at the lowest energy state is known as having kinetic stability or being kinetically persistent. {\displaystyle {\dot {x}}(t)=x} Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems.The most important type is that concerning the stability of solutions near to a point of equilibrium. ) , Suppose stable Universe, whose quantum vacuum will never decay. This latter condition has been generalized to switched systems: a linear switched discrete time system (ruled by a set of matrices { t (There are examples showing that attractivity does not imply asymptotic stability. It has been shown [13] that near to a point of equilibrium which is Lyapunov stable the system remains stable under small disturbances. Only, that last example has a catch to it: sometimes, if your conditions arent precisely right, your ball wont end up in the lowest-energy state possible. At an equilibrium point, the slope is zero and is a stable (unstable) equilibrium = , But earlier on, the Universe underwent transitions: from higher-energy states to lower-energy ones. The fact that the Universe appears to be consistent with these presumptions at least, to the limits of our observations seems to support this view, placing great constraints on how much its possible these various aspects of reality have evolved. is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of ( where H.O.T. V ( z . Metastable states that persist for many seconds (or years) are found in energetic valleys which are not the lowest possible valley (point 1 in illustration). All aboard! A much more stable position is for the ball to be down somewhere at the bottom of the valley. should satisfy a no-flux condition, so that fluid does not leak out at the boundaries e is a function of time. This appears extremely close to the metastable/stable border, where the blue dot and the three blue circles below represent 1-sigma, 2-sigma, and 3-sigma departures from the mean value. The first of these conditions is provided by details at the boundary. R This may be discussed by the theory of Aleksandr Lyapunov.In simple terms, if the solutions that start out near an equilibrium point stay near An object precisely located at L1, L2, L3, L4, or L5 will continue to orbit the parent star with the same period as the secondary body, but only L4 and L5 are stable, and only if the mass ratio between the primary and secondary masses are large enough. Matter in a stable form minimizes the stored energy in the electric and magnetic fields within it; this defines the equilibrium positions and spacing of the atomic nuclei in a molecule or an extended solid and the form of their combined electron This gravitational effect can apply just as well to binary star systems as to star-planet systems or planet-moon systems. (2) thin rod standing vertically is slightly disturbed from its position it will not come back to its original position. = Example(1) pencil standing on its point or a stick in vertically standing position. ( 2 m {\displaystyle U(x,z)=W(x,z)=0,\,} [10] The map of which state is the most stable as a function of pressure, temperature and/or composition is known as a phase diagram. 0 Is this something we actually have to worry about? The IUPAC recommends referring to these as transient rather than metastable. {\displaystyle A} However if an input changes at the wrong moment a digital circuit which employs feedback (even a simple circuit such as a flip-flop) can enter a metastable state and take an unbounded length of time to finally settle into a fully stable digital state. {\displaystyle x_{e}} x Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium. In general, the cloud is unstable if it is either very massive at a given temperature or very cool at a given mass; under these circumstances, the gas pressure cannot overcome gravity, and the cloud will collapse. The perturbation velocities It started with a bang, but wont end with one. In chemistry and physics, metastability denotes an intermediate energetic state within a dynamical system other than the system's state of least energy. Chetaev, N. G. On stable trajectories of dynamics, Kazan Univ Sci Notes, vol.4 no.1 1936; The Stability of Motion, Originally published in Russian in 1946 by . x 2 gives The other three conditions are provided by details at the interface It was recently discovered that the fluid equations governing the linear dynamics of the system admit a parity-time symmetry, and the KelvinHelmholtzRayleighTaylor instability occurs when and only when the parity-time symmetry breaks spontaneously.[18]. where The limit of the visible Universe is 46.1 billion light-years, as thats the limit of how far away an object that emitted light that would just be reaching us today would be after expanding away from us for 13.8 billion years. = t . 0 no. u A large number of publications appeared then and since in the control and systems literature. {\displaystyle w(t)} This concept is depicted in the figure, where it is observed that the two counter-rotating vortices have velocity fields that sum at the peak and trough of the perturbed interface. The Potential Energy, in this case, is given by Using the equation for Force (F) above, we have-This is generally all we use to solve a spring question. = ) The most undervalued problem-solving tool? This example is from Chemistry. ( If the Potential Energy curve is like a cup, any object (like a ball) will try to get to the bottom of that cup! , the kinematic condition holds: where the velocity Transitions from metastable excited levels are typically those forbidden by electric dipole selection rules. In the classical world, you must overcome the hill or barrier confining you to the false minimum to arrive elsewhere. = . {\displaystyle y=x-\phi (t)} Continuity of vertical velocity: At Take a look at this crazy potential energy curve. Planets within stellar systems typically organize themselves into resonant, nearly-circular orbits, as their mutual gravitational influences iron out imperfections over time, sometimes at the cost of gravitationally ejecting one or more members. Because the fluid is assumed incompressible, this velocity field has the streamfunction representation, where the subscripts indicate partial derivatives. Even with the Universe expanding, and even with that expansion accelerating due to dark energy, if a vacuum decay event such as the one envisioned here occurred anywhere within 18 billion light-years of us, at present, it would eventually reach us, destroying every atom at the speed of light in a Ghostbusters-level event when it did. {notificationOpen=false}, 2000);" x-data="{notificationOpen: false, notificationTimeout: undefined, notificationText: ''}">. It is introduced as follows for a system is (locally) attractive if. ) 0. Atomic nuclei, composed of protons and neutrons, always have a most-stable configuration for any unique number of protons and neutrons composing that nucleus. The equilibrium is Lyapunov stable but not asymptotically stable. x Sandpiles are one system which can exhibit metastability if a steep slope or tunnel is present. z

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