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=== Introduction ===
=== Introduction ===
A central concept of statistical mechanics is the ensemble. An ensemble consists                 
A central concept of statistical mechanics is the ensemble. An ensemble consists                 
of a large number of virtual copies of the system of interest. An ensemble depends             
of a large number of virtual copies of the system of interest. An ensemble will always depend on three thermodynamic state variables, as for example particle number N, temperature T and pressure p.                       
on external variables. An ensemble will always depend on three controlled external             
variables, as for example particle number N, temperature T and pressure p.                       
These three variables determine the type of ensemble that is studied. Depending on               
These three variables determine the type of ensemble that is studied. Depending on               
these three variables there is a thermodynamic potential associated with the                     
these three variables there is a thermodynamic potential associated with the                     
ensemble, which would be the Helmholtz free energy in the case of N,T and p.                     
ensemble, which would be the Gibbs free energy in the case of N,T and p.                     
Therefore, the concept of the ensemble gives access to any thermodynamic quantity.               
Therefore, the concept of the ensemble gives access to any thermodynamic quantity.               
The configurations of your system building up the ensemble can be obtained from                   
The configurations of your system building up the ensemble can be obtained from                   
molecular dynamics simulations. The molecular-dynamics approach generates the configurations for  
molecular dynamics simulations. The molecular-dynamics approach generates the configurations for  
the ensemble by integrating Newton's equations of motion.                                      
the ensemble by integrating Newton's equations of motion.
 
=== Theory ===
=== Theory ===


Line 17: Line 16:
entity in statistical mechanics. As the wave function in quantum mechanics it contains             
entity in statistical mechanics. As the wave function in quantum mechanics it contains             
all the information about a statistical system. The partition function depends on three             
all the information about a statistical system. The partition function depends on three             
thermodynamic variables such as N,T and volume V.                                                   
thermodynamic state variables such as N,T and volume V.                                                   
                                                                                                    
                                                                                                    
''' Microcanonical ensemble (N,V,E)'''
''' Microcanonical ensemble (N,V,E)'''
                                                    
                                                    
To start, the three controlled external parameters have to be defined. In the case of the             
To start, the three controlled external parameters have to be defined. In the case of the             
microcanonical ensemble these are the particle number, the volume and the internal energy E.      
microcanonical ensemble these are the particle number, the volume and the total energy E of the system.
The internal energy is the sum of the kinetic energy and potential energy of the particle           
The total energy is the sum of the kinetic energy and potential energy of the particle           
system. The partition function is written as a sum over all states in agreement with these       
system. Therefore the total energy depends on the momenta and the positions of the system.
constraints
Furthermore, the energy is an extensive variable depending on the particle number and the volume of the system.
                                                                                   
The partition function is written as a sum over all microstates in agreement with the NVE constraints
''' NVT ensmble '''
 
<math>
\Omega(N,V,E) = \sum_{E-\delta E < E(N,V,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\}) < E + \delta E } 1.
</math>
 
In this equation <math> \delta E </math> denotes a infinitesimal energy. <math>\{\mathbf{r}_{i}\} </math>
denotes the set of particle positions and <math>\{\mathbf{p}_{i}\}</math> are the conjugate momenta.
The sum is over all particle positions and momenta giving an energy in agreement with the energy constraint.
Therefore, <math> \Omega(N,V,E) </math> is the number of microstates under the given conditions.
From the microcanonical partition function it is possible to obtain the entropy by
 
<math>
S(N,V,E) = k_{B}log\Omega(N,V,E),
</math>
 
where <math> k_{B}</math> is the Boltzmann constant. The probabilities for the different micro states (r) are given by
 
<math>
 
P_{r}(N,V,E) =
\begin{cases} \frac{1}{\Omega(N,V,E)}\ for\ E-\delta E < E < E + \delta E \\
              0\ else
\end{cases}
</math>.
 
The microcanonical ensemble is a theoretical realization of a N particle system which is placed in a box
with fixed volume and fully isolated from its surroundings. Therefore, no energy exchange with the rest of the universe is possible. In VASP this ensemble can be approximated by sampling the configurations in a [[NVE_ensemble|molecular-dynamics]] run.
                                                                                 
''' Canonical ensemble ensemble (N,V,T)'''
 
In the canonical ensemble the controlled thermodynamic state variables are the particle number, the volume and the temperature. The ensemble consists of all configurations accessible to the system at the given (N,V,T) conditions. The canonical partition function can be derived to be
 
<math>
Z(N,V,T) = \sum_{r}e^{-\beta E(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})} 1
</math>,


where <math> \beta </math> is the inverse temperature <math>\frac{1}{k_{B}T}</math>.
The probability for a single microstate r is


''' NpT ensmble '''
<math>
P_{r}(N,V,T) = \frac{1}{Z(N,V,T)}e^{-\beta E(N,V,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\})}
</math>.


The canonical partition function is related to the Helmholtz free energy by


''' NpH ensmble '''
<math>
F(N,V,T) = -k_{B}logZ(N,V,T)
</math>


and the average energy of the system under canonical conditions can be computed as


<math>
\langle E \rangle = -k_{B}T\frac{\partial F(N,V,T)}{\partial \beta}=
  \sum_{r}
\frac{e^{-\beta E(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})}E(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})}{Z}
                   
</math>.


The canonical ensemble can be visualized as N particle system in a fixed volume box which is allowed to exchange thermal energy with a huge heat bath surrounding it. Therefore, in equilibrium the instantaneous temperature (temperature derived from kinetic energy) is fluctuating around the heat bath temperature. Temperature fluctuations are proportional to <math>N^{-1/2}</math>. Hence, in the thermodynamic limit (<math>N \rightarrow \infty</math>) temperature fluctuations vanish. There are several techniques to realize the canonical ensemble in a computer simulation. The configurations in a NVT ensemble can be sampled from a [[NVT_ensemble|molecular-dynamics]] run.


''' Isothermal–isobaric ensemble (NpT) '''


In the isothermal–isobaric ensemble the controlled thermodynamic state variables are the particle number, the pressure p and the temperature. The ensemble consists of all configurations accessible to the system at the given (N,p,T) conditions. The isothermal–isobaric partition function can be derived to be


<math>
Y(N,p,T) = \sum_{r}e^{-\beta\left( E(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})+pV_{r}\right)}
</math>,


where <math> \beta </math> is the inverse temperature <math>\frac{1}{k_{B}T}</math> and pV
is the contribution to the energy due to volume work.
The probability for a single microstate r is
<math>
P_{r}(N,p,T) = \frac{1}{Y(Np,T)}e^{-\beta \left(E(N,p,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\})+pV_{r}\right)}
</math>.
The isothermal–isobaric partition function is related to the Gibbs free energy by
<math>
G(N,p,T) = -k_{B}logY(Np,T)
</math>
and the average internal energy of the system under isothermal–isobaric conditions can be computed as
<math>
\langle E \rangle = -k_{B}T\frac{\partial G(N,p,T)}{\partial \beta}=
  \sum_{r}
\frac{e^{-\beta \left(E(N,p,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})+pV_{r}\right)}E(N,p,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})}{Y}
                   
</math>.
The isothermal–isobaric ensemble can be visualized as N particle system in a box without rigid boundaries which is allowed to exchange thermal energy with a huge heat bath surrounding it. Because the box has no rigid sides the volume and shape will change according to the pressure difference within and outside the box. In the isothermal–isobaric ensemble both the instantaneous temperature and pressure will fluctuate around the desired temperature (heat bath) and pressure (volume reservoir), respectively. As in the canonical ensemble these fluctuations vanish in the thermodynamic limit. There are several techniques to realize the isothermal–isobaric ensemble in a computer simulation. The configurations in a NpT ensemble can be sampled from a [[NpT_ensemble|molecular-dynamics]] run.
''' Isoenthalpic–isobaric ensemble (NpH)'''
In the isoenthalpic–isobaric ensemble the controlled thermodynamic state variables are the particle number, the pressure and the enthalpy H. The ensemble consists of all configurations accessible to the system at the given (N,p,H) conditions. The isoenthalpic–isobaric partition function can be derived to be
<math>
X(N,p,H) = \sum_{H-\delta H<H(N,p,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\})<H+\delta H} 1
</math>.
The probability for a single microstate r is
<math>
P_{r}(N,p,H) =
\begin{cases} \frac{1}{X(N,p,H)}\ for\ H-\delta H < H < E + \delta H \\
              0\ else
\end{cases}
</math>.
The isoenthalpic–isobaric ensemble can be visualized as N particle system in a box without rigid boundaries which is thermally isolated from its surroundings. Because the box has no rigid sides the volume and shape will change according to the pressure difference within and outside the box. In the isoenthalpic–isobaric ensemble the instantaneous pressure will fluctuate around the desired pressure value. The configurations in a NpH ensemble can be sampled from a [[NpH_ensemble|molecular-dynamics]] run.


=== How To ===
=== How To ===
The following table gives an overview of the possible combination of ensembles and thermostats in VASP:
The following table gives an overview of the possible combinations of ensembles and thermostats in VASP:
{{Template:MDCOMBINATIONS}}
{{Template:MDCOMBINATIONS}}
[[Category:Molecular dynamics]]

Latest revision as of 06:28, 8 May 2024

Introduction

A central concept of statistical mechanics is the ensemble. An ensemble consists of a large number of virtual copies of the system of interest. An ensemble will always depend on three thermodynamic state variables, as for example particle number N, temperature T and pressure p. These three variables determine the type of ensemble that is studied. Depending on these three variables there is a thermodynamic potential associated with the ensemble, which would be the Gibbs free energy in the case of N,T and p. Therefore, the concept of the ensemble gives access to any thermodynamic quantity. The configurations of your system building up the ensemble can be obtained from molecular dynamics simulations. The molecular-dynamics approach generates the configurations for the ensemble by integrating Newton's equations of motion.

Theory

In this section various ensembles will be introduced. To describe an ensemble mathematically the partition function will be used. The partition function is the central mathematical entity in statistical mechanics. As the wave function in quantum mechanics it contains all the information about a statistical system. The partition function depends on three thermodynamic state variables such as N,T and volume V.

Microcanonical ensemble (N,V,E)

To start, the three controlled external parameters have to be defined. In the case of the microcanonical ensemble these are the particle number, the volume and the total energy E of the system. The total energy is the sum of the kinetic energy and potential energy of the particle system. Therefore the total energy depends on the momenta and the positions of the system. Furthermore, the energy is an extensive variable depending on the particle number and the volume of the system. The partition function is written as a sum over all microstates in agreement with the NVE constraints

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \Omega(N,V,E) = \sum_{E-\delta E < E(N,V,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\}) < E + \delta E } 1. }

In this equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \delta E } denotes a infinitesimal energy. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \{\mathbf{r}_{i}\} } denotes the set of particle positions and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \{\mathbf{p}_{i}\}} are the conjugate momenta. The sum is over all particle positions and momenta giving an energy in agreement with the energy constraint. Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \Omega(N,V,E) } is the number of microstates under the given conditions. From the microcanonical partition function it is possible to obtain the entropy by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle S(N,V,E) = k_{B}log\Omega(N,V,E), }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle k_{B}} is the Boltzmann constant. The probabilities for the different micro states (r) are given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle P_{r}(N,V,E) = \begin{cases} \frac{1}{\Omega(N,V,E)}\ for\ E-\delta E < E < E + \delta E \\ 0\ else \end{cases} } .

The microcanonical ensemble is a theoretical realization of a N particle system which is placed in a box with fixed volume and fully isolated from its surroundings. Therefore, no energy exchange with the rest of the universe is possible. In VASP this ensemble can be approximated by sampling the configurations in a molecular-dynamics run.

Canonical ensemble ensemble (N,V,T)

In the canonical ensemble the controlled thermodynamic state variables are the particle number, the volume and the temperature. The ensemble consists of all configurations accessible to the system at the given (N,V,T) conditions. The canonical partition function can be derived to be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle Z(N,V,T) = \sum_{r}e^{-\beta E(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})} 1 } ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \beta } is the inverse temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \frac{1}{k_{B}T}} . The probability for a single microstate r is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle P_{r}(N,V,T) = \frac{1}{Z(N,V,T)}e^{-\beta E(N,V,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\})} } .

The canonical partition function is related to the Helmholtz free energy by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle F(N,V,T) = -k_{B}logZ(N,V,T) }

and the average energy of the system under canonical conditions can be computed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \langle E \rangle = -k_{B}T\frac{\partial F(N,V,T)}{\partial \beta}= \sum_{r} \frac{e^{-\beta E(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})}E(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})}{Z} } .

The canonical ensemble can be visualized as N particle system in a fixed volume box which is allowed to exchange thermal energy with a huge heat bath surrounding it. Therefore, in equilibrium the instantaneous temperature (temperature derived from kinetic energy) is fluctuating around the heat bath temperature. Temperature fluctuations are proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle N^{-1/2}} . Hence, in the thermodynamic limit (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle N \rightarrow \infty} ) temperature fluctuations vanish. There are several techniques to realize the canonical ensemble in a computer simulation. The configurations in a NVT ensemble can be sampled from a molecular-dynamics run.

Isothermal–isobaric ensemble (NpT)

In the isothermal–isobaric ensemble the controlled thermodynamic state variables are the particle number, the pressure p and the temperature. The ensemble consists of all configurations accessible to the system at the given (N,p,T) conditions. The isothermal–isobaric partition function can be derived to be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle Y(N,p,T) = \sum_{r}e^{-\beta\left( E(N,V,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})+pV_{r}\right)} } ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \beta } is the inverse temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \frac{1}{k_{B}T}} and pV is the contribution to the energy due to volume work. The probability for a single microstate r is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle P_{r}(N,p,T) = \frac{1}{Y(Np,T)}e^{-\beta \left(E(N,p,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\})+pV_{r}\right)} } .

The isothermal–isobaric partition function is related to the Gibbs free energy by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle G(N,p,T) = -k_{B}logY(Np,T) }

and the average internal energy of the system under isothermal–isobaric conditions can be computed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \langle E \rangle = -k_{B}T\frac{\partial G(N,p,T)}{\partial \beta}= \sum_{r} \frac{e^{-\beta \left(E(N,p,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})+pV_{r}\right)}E(N,p,\{\mathbf{r}_{r}\},\{\mathbf{p}_{r}\})}{Y} } .

The isothermal–isobaric ensemble can be visualized as N particle system in a box without rigid boundaries which is allowed to exchange thermal energy with a huge heat bath surrounding it. Because the box has no rigid sides the volume and shape will change according to the pressure difference within and outside the box. In the isothermal–isobaric ensemble both the instantaneous temperature and pressure will fluctuate around the desired temperature (heat bath) and pressure (volume reservoir), respectively. As in the canonical ensemble these fluctuations vanish in the thermodynamic limit. There are several techniques to realize the isothermal–isobaric ensemble in a computer simulation. The configurations in a NpT ensemble can be sampled from a molecular-dynamics run.

Isoenthalpic–isobaric ensemble (NpH)

In the isoenthalpic–isobaric ensemble the controlled thermodynamic state variables are the particle number, the pressure and the enthalpy H. The ensemble consists of all configurations accessible to the system at the given (N,p,H) conditions. The isoenthalpic–isobaric partition function can be derived to be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle X(N,p,H) = \sum_{H-\delta H<H(N,p,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\})<H+\delta H} 1 } .

The probability for a single microstate r is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle P_{r}(N,p,H) = \begin{cases} \frac{1}{X(N,p,H)}\ for\ H-\delta H < H < E + \delta H \\ 0\ else \end{cases} } .

The isoenthalpic–isobaric ensemble can be visualized as N particle system in a box without rigid boundaries which is thermally isolated from its surroundings. Because the box has no rigid sides the volume and shape will change according to the pressure difference within and outside the box. In the isoenthalpic–isobaric ensemble the instantaneous pressure will fluctuate around the desired pressure value. The configurations in a NpH ensemble can be sampled from a molecular-dynamics run.

How To

The following table gives an overview of the possible combinations of ensembles and thermostats in VASP:

Thermostat
Ensemble Andersen Nose-Hoover Langevin Multiple Andersen
Microcanonical (NVE) MDALGO=1, ANDERSEN_PROB=0.0
Canonical (NVT) MDALGO=1 MDALGO=2 MDALGO=3 MDALGO=13
ISIF=2 ISIF=2 ISIF=2 ISIF=2
Isobaric-isothermal (NpT) not available not available MDALGO=3 not available
ISIF=3
Isoenthalpic-isobaric (NpH) MDALGO=3, ISIF=3, LANGEVIN_GAMMA=LANGEVIN_GAMMA_L=0.0

Pages in category "Ensembles"

The following 5 pages are in this category, out of 5 total.

N

P

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