Nosé-Hoover thermostat
Encyclopedia
The Nosé–Hoover thermostat is a deterministic method used in molecular dynamics
to keep the temperature around an average. It was originally introduced by Nosé and developed further by Hoover. The heat bath is made into an integral part of the system by adding an artificial variable associated with an artificial mass.
, meaning that we control the volume, the number of particles and the energy. In real life however, we control the temperature instead of the energy. Because of the nature of the simulation, it's not possible to switch from the microcanonical ensemble
to the canonical ensemble
in which we control the temperature instead of the energy. Several methods have been introduced to keep the temperature constant while using the microcanonical ensemble
. Popular techniques to control temperature include velocity rescaling (Andersen's method), the Nosé–Hoover thermostat, Nosé–Hoover chains, the Berendsen thermostat
and Langevin dynamics
. The central idea is to simulate in such a way that we obtain a canonical distribution.
with an extra degree of freedom s. The total hamiltonian we will use to simulate is,
where g is the number of independent momentum degrees of freedom of the system, R and P represent all coordinates
and 
and Q is a parameter that should be chosen carefully. The coordinates R, P and t in this hamiltonian are virtual. They are related to the real coordinates as follows:
,
where the coordinates with an accent are the real coordinates. It can be shown that when using this hamiltonian, taking a microcanonical ensemble
average is the same as a canonical ensemble
average when
.
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
to keep the temperature around an average. It was originally introduced by Nosé and developed further by Hoover. The heat bath is made into an integral part of the system by adding an artificial variable associated with an artificial mass.
Introduction
In classic molecular dynamics simulations are done in the microcanonical ensembleMicrocanonical ensemble
In statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...
, meaning that we control the volume, the number of particles and the energy. In real life however, we control the temperature instead of the energy. Because of the nature of the simulation, it's not possible to switch from the microcanonical ensemble
Microcanonical ensemble
In statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...
to the canonical ensemble
Canonical ensemble
The canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...
in which we control the temperature instead of the energy. Several methods have been introduced to keep the temperature constant while using the microcanonical ensemble
Microcanonical ensemble
In statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...
. Popular techniques to control temperature include velocity rescaling (Andersen's method), the Nosé–Hoover thermostat, Nosé–Hoover chains, the Berendsen thermostat
Berendsen thermostat
The Berendsen thermostat is an algorithm to re-scale the velocities of particles in molecular dynamics simulations to control the simulation temperature.-Basic Description:In this scheme, the system is weakly coupled to a heat bath with some temperature...
and Langevin dynamics
Langevin dynamics
In physics, Langevin dynamics is an approach to the mathematical modeling of the dynamics of molecular systems, originally developed by the French physicist Paul Langevin...
. The central idea is to simulate in such a way that we obtain a canonical distribution.
The Nosé-Hoover thermostat
In the approach of Nosé, we introduce a heatbath in the hamiltonianHamiltonian
Hamiltonian may refer toIn mathematics :* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* Hamiltonian...
with an extra degree of freedom s. The total hamiltonian we will use to simulate is,
where g is the number of independent momentum degrees of freedom of the system, R and P represent all coordinates
and and Q is a parameter that should be chosen carefully. The coordinates R, P and t in this hamiltonian are virtual. They are related to the real coordinates as follows:,where the coordinates with an accent are the real coordinates. It can be shown that when using this hamiltonian, taking a microcanonical ensemble
Microcanonical ensemble
In statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...
average is the same as a canonical ensemble
Canonical ensemble
The canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...
average when
.