T-6 - Revision history

Ros: /* Back to dynamics: quenches, and dynamical transitions */

2026-03-15T15:27:18Z

Back to dynamics: quenches, and dynamical transitions

← Older revision		Revision as of 17:27, 15 March 2026
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	<br>		<br>

	<li> '''Activation and Arrhenius law.''' ~~Indeed, when~~ the noise in the Langevin dynamics is weak (temperature is small), the dynamics ~~does not get~~ stuck in local minima ~~forever, but it does~~ for very large time. This time depends crucially on the <~~ins~~> energy barrier </~~ins~~> which separate the minimum from the other configurations (see Fig 6.1). The <ins>Arrhenius law</ins> states that the typical timescale <math> \tau</math> required to escape from a local minimum through a barrier of height <math> \Delta E </math> with thermal dynamics with inverse temperature <math> \beta </math> scales as <math>\tau \sim \tau_0 e^{-\beta \, \Delta E} </math>. Since in the spherical <math>p</math>-spin we have <math> \Delta E \sim N \; \Delta \epsilon </math>, then <math> \tau_{\rm eq}(T< T_d, N) \sim e^{N} </math>. A dynamics made of jumps from minimum to minimum through the crossing of energy barriers is called <ins> activated dynamics </ins>.		<li> '''Activation and Arrhenius law.''' Why exponential timescales? When the noise in the Langevin dynamics is weak (temperature is small), the dynamics gets stuck in local minima for very large time. This time depends crucially on the <em> energy barrier </em> which separate the minimum from the other configurations (see Fig 6.1). The <ins>Arrhenius law</ins> states that the typical timescale <math> \tau</math> required to escape from a local minimum through a barrier of height <math> \Delta E </math> with thermal dynamics with inverse temperature <math> \beta </math> scales as <math>\tau \sim \tau_0 e^{-\beta \, \Delta E} </math>. Since in the spherical <math>p</math>-spin we have <math> \Delta E \sim N \; \Delta \epsilon </math>, then <math> \tau_{\rm eq}(T< T_d, N)> \tau_0 e^{-\beta \, \Delta E}\sim e^{N} </math>. A dynamics made of jumps from minimum to minimum through the crossing of energy barriers is called <ins> activated dynamics </ins>.
	</li>		</li>
	<br>		<br>

Ros: /* Back to dynamics: quenches, and dynamical transitions */

2026-03-15T15:25:31Z

Back to dynamics: quenches, and dynamical transitions

← Older revision		Revision as of 17:25, 15 March 2026
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	[[File:Activated Jump.png\|thumb\|right\|x160px\|Fig. 6 - Activated jump across an energy barrier.]]		[[File:Activated Jump.png\|thumb\|right\|x160px\|Fig. 6 - Activated jump across an energy barrier.]]

	<li> '''Equilibration timescales.''' Does it mean that when <math>T<T_d</math>, the system <em>never</em> equilibrates? This is true only in the limit <math>N \to \infty</math>. When <math>N </math> is finite, there is a timescale <math>\tau_{\rm eq}(T, N)</math> beyond which the system equilibrates. ~~So, the dynamical transition is a transition only for <math>N \to \infty</math>, and a crossover for finite <math>N</math>.~~ However, this equilibration timescale		<li> '''Equilibration timescales.''' Does it mean that when <math>T<T_d</math>, the system <em>never</em> equilibrates? This is true only in the limit <math>N \to \infty</math>. When <math>N </math> is finite, there is a timescale <math>\tau_{\rm eq}(T, N)</math> beyond which the system equilibrates. However, this equilibration timescale
	in the spherical <math>p</math>-spin scales as		in the spherical <math>p</math>-spin scales as
	<math display="block">		<math display="block">
	\tau_{\rm eq}(T< T_d, N) \sim e^{N}.		\tau_{\rm eq}(T< T_d, N) \sim e^{N}.
	</math>		</math>
	This is again due to the presence of many local minima/ metastable states, that are separated by <ins>extensive</ins> energy barriers.		This is again due to the presence of many local minima/ metastable states, that are separated by <ins>extensive</ins> energy barriers. So, when we take <math>N \to \infty</math> before taking the large time limit, we are unable to see equilibration and we have a sharp transition, which becomes a crossover for finite <math>N</math>.
	</li>		</li>
	<br>		<br>

Ros: /* Back to dynamics: quenches, and dynamical transitions */

2026-03-15T15:23:55Z

Back to dynamics: quenches, and dynamical transitions

← Older revision		Revision as of 17:23, 15 March 2026
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	<li> '''Gradient descent dynamics.''' The local minima are dynamically stable: if we do gradient descent, we get stuck in a local minimum and we exert a small perturbation to the configuration, gradient descent brings us back to the local minimum. These configurations are <~~ins~~>trapping</~~ins~~>. If we try to optimize the landscape, i.e. to reach the ground state, with gradient descent dynamics, we expect that we will not be able to reach it easily, as we will be trapped by local minima. In fact, for the spherical <math>p</math>-spin model it can be shown that starting from random initial conditions and evolving the configuration with gradient descent (possibly with infinitesimal noise, to be sent to zero with a protocol),		<li> '''Gradient descent dynamics.''' The local minima are dynamically stable: if we do gradient descent, we get stuck in a local minimum and we exert a small perturbation to the configuration, gradient descent brings us back to the local minimum. These configurations are <em>trapping</em>. If we try to optimize the landscape, i.e. to reach the ground state, with gradient descent dynamics, we expect that we will not be able to reach it easily, as we will be trapped by local minima. In fact, for the spherical <math>p</math>-spin model it can be shown that starting from random initial conditions and evolving the configuration with gradient descent (possibly with infinitesimal noise, to be sent to zero with a protocol),
	<math display="block">		<math display="block">
	\lim_{t \to \infty} \lim_{N \to \infty} \frac{ E(\vec{\sigma}(t))}{N} = \epsilon_{\rm th} \neq \epsilon_{\rm gs}.		\lim_{t \to \infty} \lim_{N \to \infty} \frac{ E(\vec{\sigma}(t))}{N} = \epsilon_{\rm th} \neq \epsilon_{\rm gs}.
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	[[File:Activated Jump.png\|thumb\|right\|x160px\|Fig. 6 - Activated jump across an energy barrier.]]		[[File:Activated Jump.png\|thumb\|right\|x160px\|Fig. 6 - Activated jump across an energy barrier.]]

	<li> '''Equilibration timescales.''' Does it mean that when <math>T<T_d</math>, the system <~~emph~~>never</~~emph~~> equilibrates? This is true only in the limit <math>N \to \infty</math>. When <math>N </math> is finite, there is a timescale <math>\tau_{\rm eq}(T, N)</math> beyond which the system equilibrates. So, the dynamical transition is a transition only for <math>N \to \infty</math>, and a crossover for finite <math>N</math>. However, this equilibration timescale		<li> '''Equilibration timescales.''' Does it mean that when <math>T<T_d</math>, the system <em>never</em> equilibrates? This is true only in the limit <math>N \to \infty</math>. When <math>N </math> is finite, there is a timescale <math>\tau_{\rm eq}(T, N)</math> beyond which the system equilibrates. So, the dynamical transition is a transition only for <math>N \to \infty</math>, and a crossover for finite <math>N</math>. However, this equilibration timescale
	in the spherical <math>p</math>-spin scales as		in the spherical <math>p</math>-spin scales as
	<math display="block">		<math display="block">

Ros: /* Back to dynamics: quenches, and dynamical transitions */

2026-03-15T15:22:50Z

Back to dynamics: quenches, and dynamical transitions

← Older revision		Revision as of 17:22, 15 March 2026
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	\lim_{t \to \infty} \lim_{N \to \infty} \frac{E(\vec{\sigma}(t))}{N} \neq \epsilon_{\rm eq}(T).		\lim_{t \to \infty} \lim_{N \to \infty} \frac{E(\vec{\sigma}(t))}{N} \neq \epsilon_{\rm eq}(T).
	</math>		</math>
	~~This~~ is called the <ins>dynamical transition temperature</ins>.		The statement above for gradient descent corresponds to the special case <math>T=0</math>. <math>T_d</math> is called the <ins>dynamical transition temperature</ins>.
	</li>		</li>
	<br>		<br>

Ros: /* Back to dynamics: quenches, and dynamical transitions */

2026-03-15T15:21:28Z

Back to dynamics: quenches, and dynamical transitions

← Older revision		Revision as of 17:21, 15 March 2026
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	<li> '''Dynamical transition.''' Now, in the spherical <math>p</math>-spin we know that if <math>\epsilon_{\rm eq}(T)>\epsilon_{\rm th}</math>, the energy shell has many stationary points, but they are all unstable saddles and do not trap the dynamics. We expect that this energy shell is relatively easy to explore dynamically, and that equilibration takes place. On the other hand, if <math>\epsilon_{\rm eq}(T)<\epsilon_{\rm th}</math>, in the equilibrium energy shell and at higher energy, there are exponentially many local minima that trap the dynamics, and we expect that reaching equilibrium configurations will be difficult. This tells us that there exists a critical <math>T_d</math>, defined by		<li> '''Dynamical transition.''' Now, in the spherical <math>p</math>-spin we know that if <math>\epsilon_{\rm eq}(T)>\epsilon_{\rm th}</math>, the energy shell has many stationary points, but they are all unstable saddles and do not trap the dynamics. We expect that this energy shell is relatively easy to explore dynamically, and that equilibration takes place. On the other hand, if <math>\epsilon_{\rm eq}(T)<\epsilon_{\rm th}</math>, in the equilibrium energy shell and at higher energy, there are exponentially many local minima that trap the dynamics, and we expect that reaching equilibrium configurations will be difficult. This tells us that there exists a critical <math>T_d</math>, defined by
	<math display="block">		<math display="block">
	\epsilon_{\rm eq}(T_d)=\epsilon_{\rm th},		\epsilon_{\rm eq}(T_d)=\epsilon_{\rm th},

Ros: /* Back to dynamics: quenches, and dynamical transitions */

2026-03-15T15:20:53Z

Back to dynamics: quenches, and dynamical transitions

← Older revision		Revision as of 17:20, 15 March 2026
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	<li> '''Quenches.''' We can generalize this protocol to higher <math>T</math>: we extract randomly the initial condition of the dynamics, and then we evolve the configuration with Langevin dynamics (gradient descent + noise):		<li> '''Quenches in temperature and equilibration.''' We can generalize this protocol to higher <math>T</math>: we extract randomly the initial condition of the dynamics, and then we evolve the configuration with Langevin dynamics (gradient descent + noise):
	<math display="block">		<math display="block">
	\frac{d \vec{\sigma}(t)}{dt}=- \~~nabla_~~\perp E(\vec{\sigma})+ \vec{\eta}_\perp(t), \quad \quad \langle \eta_i(t) \eta_j(t')\rangle= 2 T \delta_{ij} \delta(t-t')		\frac{d \vec{\sigma}(t)}{dt}=- {\nabla}_\perp E(\vec{\sigma})+ {\vec{\eta}}_\perp(t), \quad \quad \langle \eta_i(t) \eta_j(t')\rangle= 2 T \delta_{ij} \delta(t-t')
	</math>		</math>
	In Langevin dynamics, <math>\vec{\eta}_\perp(t)</math> a Gaussian vector at each time <math> t </math>, uncorrelated from the vectors at other times <math> t' \neq t </math>, with zero average and ~~some~~ constant variance proportional to temperature. ~~They represent effectively~~ the action of a bath on the system.		In Langevin dynamics, <math>{\vec{\eta}}_\perp(t)</math> a Gaussian vector at each time <math> t </math>, uncorrelated from the vectors at other times <math> t' \neq t </math>, with zero average and constant variance proportional to temperature. It represents the action of a thermal bath on the system.
	This dynamical protocol is called a ~~temperature~~ quench. The question we can ask is: does the system equilibrate with the bath under this dynamics? If yes, we should see that		This dynamical protocol is called a <ins>quench </ins>. The question we can ask is: does the system equilibrate with the bath under this dynamics? If yes, we should see that
	<math display="block">		<math display="block">
	\lim_{t \to \infty} \lim_{N \to \infty} \frac{E(\vec{\sigma}(t))}{N} = \epsilon_{\rm eq}(T),		\lim_{t \to \infty} \lim_{N \to \infty} \frac{E(\vec{\sigma}(t))}{N} = \epsilon_{\rm eq}(T),
	</math>		</math>
	where <math>\epsilon_{\rm eq}(T)</math> is the equilibrium energy density ~~(that we can compute using Boltzmann)~~. Equilibrating with the bath would indeed imply that at large time the system visits uniformly the equilibrium energy shell.		where <math>\epsilon_{\rm eq}(T)</math> is the equilibrium energy density at the temperature <math> T </math>, the same one controlling the strength of the noise. Equilibrating with the bath would indeed imply that at large time the system visits uniformly the equilibrium energy shell.
	</li>		</li>
	<br>		<br>

Ros: /* Back to dynamics: quenches, and dynamical transitions */

2026-03-15T15:18:34Z

Back to dynamics: quenches, and dynamical transitions

← Older revision		Revision as of 17:18, 15 March 2026
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	<li> '''Gradient descent dynamics.''' The local minima are dynamically stable: if I do gradient descent, I get stuck in a local minimum and I exert a small perturbation to the configuration, gradient descent brings me back to the local minimum. ~~In this sense, these~~ configurations are trapping. ~~Therefore, if I~~ try to optimize the landscape, i.e. to reach the ground state, with gradient descent dynamics, I expect that I will not be able to reach ~~the ground state~~ easily, as I will be trapped by ~~these~~ local minima. In fact, for the spherical <math>p</math>-spin model it can be shown that starting ~~the gradient descent dynamics~~ from random initial conditions and evolving the configuration with gradient descent (possibly with infinitesimal noise),		<li> '''Gradient descent dynamics.''' The local minima are dynamically stable: if we do gradient descent, we get stuck in a local minimum and we exert a small perturbation to the configuration, gradient descent brings us back to the local minimum. These configurations are <ins>trapping</ins>. If we try to optimize the landscape, i.e. to reach the ground state, with gradient descent dynamics, we expect that we will not be able to reach it easily, as we will be trapped by local minima. In fact, for the spherical <math>p</math>-spin model it can be shown that starting from random initial conditions and evolving the configuration with gradient descent (possibly with infinitesimal noise, to be sent to zero with a protocol),
	<math display="block">		<math display="block">
	\lim_{t \to \infty} \lim_{N \to \infty} \frac{ E(\vec{\sigma}(t))}{N} = \epsilon_{\rm th} \neq \epsilon_{\rm gs}.		\lim_{t \to \infty} \lim_{N \to \infty} \frac{ E(\vec{\sigma}(t))}{N} = \epsilon_{\rm th} \neq \epsilon_{\rm gs}.

Ros: /* Back to dynamics: quenches, and dynamical transitions */

2026-03-15T15:16:50Z

Back to dynamics: quenches, and dynamical transitions

← Older revision		Revision as of 17:16, 15 March 2026
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	== Back to dynamics: quenches, and dynamical transitions ==		== Back to dynamics: quenches, and dynamical transitions ==
	~~[[File:Activated Jump.png\|thumb\|right\|x160px\|Fig 7 - Activated jump across an energy barrier.]]~~

	Through Problems 5 and 6, we have shown that the energy landscape of the spherical <math>p</math>-spin model has exponentially many stationary points , and that there is a transition at the energy density <math>\epsilon_{\rm th}</math>: for <math>\epsilon>\epsilon_{\rm th}</math> the stationary points are saddles, for <math>\epsilon\leq \epsilon_{\rm th}</math> they are local minima. Let us try to deduce something on the systems's dynamics out of this.		Through Problems 5 and 6, we have shown that the energy landscape of the spherical <math>p</math>-spin model has exponentially many stationary points , and that there is a transition at the energy density <math>\epsilon_{\rm th}</math>: for <math>\epsilon>\epsilon_{\rm th}</math> the stationary points are saddles, for <math>\epsilon\leq \epsilon_{\rm th}</math> they are local minima. Let us try to deduce something on the systems's dynamics out of this.
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	</li>		</li>
	<br>		<br>

			[[File:Activated Jump.png\|thumb\|right\|x160px\|Fig. 6 - Activated jump across an energy barrier.]]

	<li> '''Equilibration timescales.''' Does it mean that when <math>T<T_d</math>, the system <emph>never</emph> equilibrates? This is true only in the limit <math>N \to \infty</math>. When <math>N </math> is finite, there is a timescale <math>\tau_{\rm eq}(T, N)</math> beyond which the system equilibrates. So, the dynamical transition is a transition only for <math>N \to \infty</math>, and a crossover for finite <math>N</math>. However, this equilibration timescale		<li> '''Equilibration timescales.''' Does it mean that when <math>T<T_d</math>, the system <emph>never</emph> equilibrates? This is true only in the limit <math>N \to \infty</math>. When <math>N </math> is finite, there is a timescale <math>\tau_{\rm eq}(T, N)</math> beyond which the system equilibrates. So, the dynamical transition is a transition only for <math>N \to \infty</math>, and a crossover for finite <math>N</math>. However, this equilibration timescale

Ros: /* Back to dynamics: quenches, and dynamical transitions */

2026-03-15T15:16:19Z

Back to dynamics: quenches, and dynamical transitions

← Older revision		Revision as of 17:16, 15 March 2026
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	== Back to dynamics: quenches, and dynamical transitions ==		== Back to dynamics: quenches, and dynamical transitions ==
	[[File:~~Landscapes-GDD~~.png\|thumb\|right\|~~x200px~~\|~~Non~~-~~rugged vs rugged~~ energy ~~landscapes~~.]]		[[File:Activated Jump.png\|thumb\|right\|x160px\|Fig 7 - Activated jump across an energy barrier.]]

	Through Problems 5 and 6, we have shown that the energy landscape of the spherical <math>p</math>-spin model has exponentially many stationary points , and that there is a transition at the energy density <math>\epsilon_{\rm th}</math>: for <math>\epsilon>\epsilon_{\rm th}</math> the stationary points are saddles, for <math>\epsilon\leq \epsilon_{\rm th}</math> they are local minima. Let us try to deduce something on the systems's dynamics out of this.		Through Problems 5 and 6, we have shown that the energy landscape of the spherical <math>p</math>-spin model has exponentially many stationary points , and that there is a transition at the energy density <math>\epsilon_{\rm th}</math>: for <math>\epsilon>\epsilon_{\rm th}</math> the stationary points are saddles, for <math>\epsilon\leq \epsilon_{\rm th}</math> they are local minima. Let us try to deduce something on the systems's dynamics out of this.

Ros: /* Back to dynamics: quenches, and dynamical transitions */

2026-03-15T15:15:25Z

Back to dynamics: quenches, and dynamical transitions

← Older revision		Revision as of 17:15, 15 March 2026
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	\lim_{t \to \infty} \lim_{N \to \infty} \frac{E(\vec{\sigma}(t))}{N} \neq \epsilon_{\rm eq}(T).		\lim_{t \to \infty} \lim_{N \to \infty} \frac{E(\vec{\sigma}(t))}{N} \neq \epsilon_{\rm eq}(T).
	</math>		</math>
	This is called the <~~emph~~>dynamical transition temperature</~~emph~~>.		This is called the <ins>dynamical transition temperature</ins>.
	</li>		</li>
	<br>		<br>
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	\tau_{\rm eq}(T< T_d, N) \sim e^{N}.		\tau_{\rm eq}(T< T_d, N) \sim e^{N}.
	</math>		</math>
	This is again due to the presence of many local minima/ metastable states, that are separated by <~~emph~~>extensive</~~emph~~> energy barriers.		This is again due to the presence of many local minima/ metastable states, that are separated by <ins>extensive</ins> energy barriers.
			</li>
			<br>

			<li> '''Activation and Arrhenius law.''' Indeed, when the noise in the Langevin dynamics is weak (temperature is small), the dynamics does not get stuck in local minima forever, but it does for very large time. This time depends crucially on the <ins> energy barrier </ins> which separate the minimum from the other configurations (see Fig 6.1). The <ins>Arrhenius law</ins> states that the typical timescale <math> \tau</math> required to escape from a local minimum through a barrier of height <math> \Delta E </math> with thermal dynamics with inverse temperature <math> \beta </math> scales as <math>\tau \sim \tau_0 e^{-\beta \, \Delta E} </math>. Since in the spherical <math>p</math>-spin we have <math> \Delta E \sim N \; \Delta \epsilon </math>, then <math> \tau_{\rm eq}(T< T_d, N) \sim e^{N} </math>. A dynamics made of jumps from minimum to minimum through the crossing of energy barriers is called <ins> activated dynamics </ins>.
	<~~sup~~>~~[[#Notes\|[*] ]]~~</~~sup~~>.		</li>

	<~~div style="font~~-~~size:89%"~~>
	: <~~small~~>~~[*]~~</~~small~~> - ~~This quantity looks similar to the entropy~~ <math> S(\epsilon) </math> ~~we computed for the REM in Problem 1. However~~, ~~while the entropy counts all configurations at a given energy density, the complexity~~ <math> \~~Sigma~~(\~~epsilon)~~ </math> ~~accounts only for~~ the ~~stationary points~~.
	</~~div~~>
	<br>		<br>