# Is there a mathematically precise definition of metastability?

by Tobias Fritz
Tags: definition, mathematically, metastability, precise
 P: n/a In statistical physics, metastability seems to be an important concept. Though it seems kind of vague to me so far. Is there a mathematically precise definition, say when an arbitrary dynamical system is in a metastable state? TIA, Tobias
 P: n/a Tobias Fritz wrote: > In statistical physics, metastability seems to be an important concept. > Though it seems kind of vague to me so far. Is there a mathematically > precise definition, say when an arbitrary dynamical system is in a > metastable state? The difference between a stable and a metastable state is the same as between a global and a local minimum. A metastable state describes a local minimum in the energy of a system (usually a thermodynamic system). Since it is a local minimum, small peturbations will not push it away from this state, at least for some time. However, given enough time (due to fluctuations, tunneling, dissipation, etc.) the system will move into a stable state. Stable states are globally stable, and are defined by global minima of the system's energy. Practically speaking, there is no difference between a very long lived metastable state and a stable one. For example, the Bose-Einstein condensates that have recently become experimentally realizable in cold atomic gases are actually metastable with lifetimes on the order of minutes (my memory may not be 100% correct on this). Which in any case is orders of magnitude greater than the timescale characteristic of atomic interactions. Another note. First order phase transitions (where we see one phase discontinuously jump into another one, like supercooled water turning to ice) indicate the presense of metastable states. Under controlled circumstances, the metastable state may persist even though it is not globally energetically favorable. However, under a large enough perturbation it will change to the more energetically favorable state. Hope this helps. Igor
 P: n/a Tobias Fritz wrote: > In statistical physics, metastability seems to be an important concept. > Though it seems kind of vague to me so far. Is there a mathematically > precise definition, say when an arbitrary dynamical system is in a > metastable state? The difference between a stable and a metastable state is the same as between a global and a local minimum. A metastable state describes a local minimum in the energy of a system (usually a thermodynamic system). Since it is a local minimum, small peturbations will not push it away from this state, at least for some time. However, given enough time (due to fluctuations, tunneling, dissipation, etc.) the system will move into a stable state. Stable states are globally stable, and are defined by global minima of the system's energy. Practically speaking, there is no difference between a very long lived metastable state and a stable one. For example, the Bose-Einstein condensates that have recently become experimentally realizable in cold atomic gases are actually metastable with lifetimes on the order of minutes (my memory may not be 100% correct on this). Which in any case is orders of magnitude greater than the timescale characteristic of atomic interactions. Another note. First order phase transitions (where we see one phase discontinuously jump into another one, like supercooled water turning to ice) indicate the presense of metastable states. Under controlled circumstances, the metastable state may persist even though it is not globally energetically favorable. However, under a large enough perturbation it will change to the more energetically favorable state. Hope this helps. Igor
P: n/a

## Is there a mathematically precise definition of metastability?

Tobias Fritz wrote:
> In statistical physics, metastability seems to be an important concept.
> Though it seems kind of vague to me so far. Is there a mathematically
> precise definition, say when an arbitrary dynamical system is in a
> metastable state?

The difference between a stable and a metastable state is the same as
between a global and a local minimum. A metastable state describes a
local minimum in the energy of a system (usually a thermodynamic
system). Since it is a local minimum, small peturbations will not push
it away from this state, at least for some time. However, given enough
time (due to fluctuations, tunneling, dissipation, etc.) the system
will move into a stable state. Stable states are globally stable, and
are defined by global minima of the system's energy.

Practically speaking, there is no difference between a very long lived
metastable state and a stable one. For example, the Bose-Einstein
condensates that have recently become experimentally realizable in cold
atomic gases are actually metastable with lifetimes on the order of
minutes (my memory may not be 100% correct on this). Which in any case
is orders of magnitude greater than the timescale characteristic of
atomic interactions.

Another note. First order phase transitions (where we see one phase
discontinuously jump into another one, like supercooled water turning
to ice) indicate the presense of metastable states. Under controlled
circumstances, the metastable state may persist even though it is not
globally energetically favorable. However, under a large enough
perturbation it will change to the more energetically favorable state.

Hope this helps.

Igor

 P: n/a Tobias Fritz wrote: > In statistical physics, metastability seems to be an important concept. > Though it seems kind of vague to me so far. Is there a mathematically > precise definition, say when an arbitrary dynamical system is in a > metastable state? The difference between a stable and a metastable state is the same as between a global and a local minimum. A metastable state describes a local minimum in the energy of a system (usually a thermodynamic system). Since it is a local minimum, small peturbations will not push it away from this state, at least for some time. However, given enough time (due to fluctuations, tunneling, dissipation, etc.) the system will move into a stable state. Stable states are globally stable, and are defined by global minima of the system's energy. Practically speaking, there is no difference between a very long lived metastable state and a stable one. For example, the Bose-Einstein condensates that have recently become experimentally realizable in cold atomic gases are actually metastable with lifetimes on the order of minutes (my memory may not be 100% correct on this). Which in any case is orders of magnitude greater than the timescale characteristic of atomic interactions. Another note. First order phase transitions (where we see one phase discontinuously jump into another one, like supercooled water turning to ice) indicate the presense of metastable states. Under controlled circumstances, the metastable state may persist even though it is not globally energetically favorable. However, under a large enough perturbation it will change to the more energetically favorable state. Hope this helps. Igor
 P: n/a Tobias Fritz wrote: > In statistical physics, metastability seems to be an important concept. > Though it seems kind of vague to me so far. Is there a mathematically > precise definition, say when an arbitrary dynamical system is in a > metastable state? The difference between a stable and a metastable state is the same as between a global and a local minimum. A metastable state describes a local minimum in the energy of a system (usually a thermodynamic system). Since it is a local minimum, small peturbations will not push it away from this state, at least for some time. However, given enough time (due to fluctuations, tunneling, dissipation, etc.) the system will move into a stable state. Stable states are globally stable, and are defined by global minima of the system's energy. Practically speaking, there is no difference between a very long lived metastable state and a stable one. For example, the Bose-Einstein condensates that have recently become experimentally realizable in cold atomic gases are actually metastable with lifetimes on the order of minutes (my memory may not be 100% correct on this). Which in any case is orders of magnitude greater than the timescale characteristic of atomic interactions. Another note. First order phase transitions (where we see one phase discontinuously jump into another one, like supercooled water turning to ice) indicate the presense of metastable states. Under controlled circumstances, the metastable state may persist even though it is not globally energetically favorable. However, under a large enough perturbation it will change to the more energetically favorable state. Hope this helps. Igor
 P: n/a Tobias Fritz wrote: > In statistical physics, metastability seems to be an important concept. > Though it seems kind of vague to me so far. Is there a mathematically > precise definition, say when an arbitrary dynamical system is in a > metastable state? The difference between a stable and a metastable state is the same as between a global and a local minimum. A metastable state describes a local minimum in the energy of a system (usually a thermodynamic system). Since it is a local minimum, small peturbations will not push it away from this state, at least for some time. However, given enough time (due to fluctuations, tunneling, dissipation, etc.) the system will move into a stable state. Stable states are globally stable, and are defined by global minima of the system's energy. Practically speaking, there is no difference between a very long lived metastable state and a stable one. For example, the Bose-Einstein condensates that have recently become experimentally realizable in cold atomic gases are actually metastable with lifetimes on the order of minutes (my memory may not be 100% correct on this). Which in any case is orders of magnitude greater than the timescale characteristic of atomic interactions. Another note. First order phase transitions (where we see one phase discontinuously jump into another one, like supercooled water turning to ice) indicate the presense of metastable states. Under controlled circumstances, the metastable state may persist even though it is not globally energetically favorable. However, under a large enough perturbation it will change to the more energetically favorable state. Hope this helps. Igor
 P: n/a Tobias Fritz wrote: > In statistical physics, metastability seems to be an important concept. > Though it seems kind of vague to me so far. Is there a mathematically > precise definition, say when an arbitrary dynamical system is in a > metastable state? The difference between a stable and a metastable state is the same as between a global and a local minimum. A metastable state describes a local minimum in the energy of a system (usually a thermodynamic system). Since it is a local minimum, small peturbations will not push it away from this state, at least for some time. However, given enough time (due to fluctuations, tunneling, dissipation, etc.) the system will move into a stable state. Stable states are globally stable, and are defined by global minima of the system's energy. Practically speaking, there is no difference between a very long lived metastable state and a stable one. For example, the Bose-Einstein condensates that have recently become experimentally realizable in cold atomic gases are actually metastable with lifetimes on the order of minutes (my memory may not be 100% correct on this). Which in any case is orders of magnitude greater than the timescale characteristic of atomic interactions. Another note. First order phase transitions (where we see one phase discontinuously jump into another one, like supercooled water turning to ice) indicate the presense of metastable states. Under controlled circumstances, the metastable state may persist even though it is not globally energetically favorable. However, under a large enough perturbation it will change to the more energetically favorable state. Hope this helps. Igor
 P: n/a Tobias Fritz wrote: > In statistical physics, metastability seems to be an important concept. > Though it seems kind of vague to me so far. Is there a mathematically > precise definition, say when an arbitrary dynamical system is in a > metastable state? The difference between a stable and a metastable state is the same as between a global and a local minimum. A metastable state describes a local minimum in the energy of a system (usually a thermodynamic system). Since it is a local minimum, small peturbations will not push it away from this state, at least for some time. However, given enough time (due to fluctuations, tunneling, dissipation, etc.) the system will move into a stable state. Stable states are globally stable, and are defined by global minima of the system's energy. Practically speaking, there is no difference between a very long lived metastable state and a stable one. For example, the Bose-Einstein condensates that have recently become experimentally realizable in cold atomic gases are actually metastable with lifetimes on the order of minutes (my memory may not be 100% correct on this). Which in any case is orders of magnitude greater than the timescale characteristic of atomic interactions. Another note. First order phase transitions (where we see one phase discontinuously jump into another one, like supercooled water turning to ice) indicate the presense of metastable states. Under controlled circumstances, the metastable state may persist even though it is not globally energetically favorable. However, under a large enough perturbation it will change to the more energetically favorable state. Hope this helps. Igor
 P: n/a Tobias Fritz wrote: > In statistical physics, metastability seems to be an important concept. > Though it seems kind of vague to me so far. Is there a mathematically > precise definition, say when an arbitrary dynamical system is in a > metastable state? The difference between a stable and a metastable state is the same as between a global and a local minimum. A metastable state describes a local minimum in the energy of a system (usually a thermodynamic system). Since it is a local minimum, small peturbations will not push it away from this state, at least for some time. However, given enough time (due to fluctuations, tunneling, dissipation, etc.) the system will move into a stable state. Stable states are globally stable, and are defined by global minima of the system's energy. Practically speaking, there is no difference between a very long lived metastable state and a stable one. For example, the Bose-Einstein condensates that have recently become experimentally realizable in cold atomic gases are actually metastable with lifetimes on the order of minutes (my memory may not be 100% correct on this). Which in any case is orders of magnitude greater than the timescale characteristic of atomic interactions. Another note. First order phase transitions (where we see one phase discontinuously jump into another one, like supercooled water turning to ice) indicate the presense of metastable states. Under controlled circumstances, the metastable state may persist even though it is not globally energetically favorable. However, under a large enough perturbation it will change to the more energetically favorable state. Hope this helps. Igor
 P: n/a I believe a good (but physically vague :-) illustration is that most of what we construct is metastable. A car is metastable and a pile of rust is stable. Which I believe leads to a mechanism of added stability for some metastable states: Failure or breakdown mechanisms (for example in semiconductor devices) may follow an Arrhenius equation due to a potential barrier.
 P: n/a I believe a good (but physically vague :-) illustration is that most of what we construct is metastable. A car is metastable and a pile of rust is stable. Which I believe leads to a mechanism of added stability for some metastable states: Failure or breakdown mechanisms (for example in semiconductor devices) may follow an Arrhenius equation due to a potential barrier.
 P: n/a I believe a good (but physically vague :-) illustration is that most of what we construct is metastable. A car is metastable and a pile of rust is stable. Which I believe leads to a mechanism of added stability for some metastable states: Failure or breakdown mechanisms (for example in semiconductor devices) may follow an Arrhenius equation due to a potential barrier.
 P: n/a I believe a good (but physically vague :-) illustration is that most of what we construct is metastable. A car is metastable and a pile of rust is stable. Which I believe leads to a mechanism of added stability for some metastable states: Failure or breakdown mechanisms (for example in semiconductor devices) may follow an Arrhenius equation due to a potential barrier.
 P: n/a I believe a good (but physically vague :-) illustration is that most of what we construct is metastable. A car is metastable and a pile of rust is stable. Which I believe leads to a mechanism of added stability for some metastable states: Failure or breakdown mechanisms (for example in semiconductor devices) may follow an Arrhenius equation due to a potential barrier.
 P: n/a I believe a good (but physically vague :-) illustration is that most of what we construct is metastable. A car is metastable and a pile of rust is stable. Which I believe leads to a mechanism of added stability for some metastable states: Failure or breakdown mechanisms (for example in semiconductor devices) may follow an Arrhenius equation due to a potential barrier.
 P: n/a I believe a good (but physically vague :-) illustration is that most of what we construct is metastable. A car is metastable and a pile of rust is stable. Which I believe leads to a mechanism of added stability for some metastable states: Failure or breakdown mechanisms (for example in semiconductor devices) may follow an Arrhenius equation due to a potential barrier.
 P: n/a I believe a good (but physically vague :-) illustration is that most of what we construct is metastable. A car is metastable and a pile of rust is stable. Which I believe leads to a mechanism of added stability for some metastable states: Failure or breakdown mechanisms (for example in semiconductor devices) may follow an Arrhenius equation due to a potential barrier.

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