Download Understanding Molecular Simulations

Understanding
Molecular Simulations
Molsim 2012
Why Molecular Simulations
Molsim 2012
Why Molecular Simulations
Paul Dirac, after completing his formalism of quantum mechanics:
Molsim 2012
Why Molecular Simulations
Paul Dirac, after completing his formalism of quantum mechanics:
“The rest is chemistry…”.
Molsim 2012
Why Molecular Simulations
Paul Dirac, after completing his formalism of quantum mechanics:
“The rest is chemistry…”.
This is a heavy burden the shoulders of “chemistry”:
Molsim 2012
Why Molecular Simulations
Paul Dirac, after completing his formalism of quantum mechanics:
“The rest is chemistry…”.
This is a heavy burden the shoulders of “chemistry”:
The “rest” amounts to the quantitative description of the world around us
and the prediction of all every-day phenomena ranging from the
chemical reactions of small molecules to the integrated description of
living organisms.
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Concept of Molecular Simulations
for a given intermolecular potential
“exactly” predict the thermodynamic and
transport properties of the system
Molsim 2012
Concept of Molecular Simulations
for a given intermolecular potential
“exactly” predict the thermodynamic and
transport properties of the system
We assume the
interactions between
the particles are known!
Molsim 2012
Concept of Molecular
Simulations
Exact = in the limit
of
infinitely long simulations the
error bars can be made
infinitely small
for a given intermolecular potential
“exactly” predict the thermodynamic and
transport properties of the system
Molsim 2012
Concept of Molecular Simulations
for a given intermolecular potential
“exactly” predict the thermodynamic and
transport properties of the system
Pressure
Heat capacity
Heat of adsorption
Structure
….
Molsim 2012
Concept of Molecular Simulations
for a given intermolecular potential
“exactly” predict the thermodynamic and
transport properties of the system
Diffusion coefficient
Viscosity
…
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Concept of Molecular Simulations
If one could envision
an experimental
system of these
particles that interact
potential
with the potential.
for a given intermolecular
“exactly” predict the thermodynamic and
transport properties of the system
Molsim 2012
Concept of Molecular Simulations
for a given intermolecular potential
“exactly” predict the thermodynamic and
transport properties of the system
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THE question:
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“Can we predict the
macroscopic properties of
(classical) many-body systems?”
THE question:
“Can we predict the
macroscopic properties of
(classical) many-body systems?”
NEWTON says: yes, my F=ma gives
the future time evolution of the
system
Molsim 2012
THE question:
“Can we predict the
macroscopic properties of
(classical) many-body systems?”
NEWTON says: yes, my F=ma gives
the future time evolution of the
system
LAPLACE says: in principle yes,
provided that we know the
position, velocity and interaction of
all molecules, then the future
behavior is predictable,..
Molsim 2012
THE question:
“Can we predict the
macroscopic properties of
(classical) many-body systems?”
NEWTON says: yes, my F=ma gives
the future time evolution of the
system
LAPLACE says: in principle yes,
provided that we know the
position, velocity and interaction of
all molecules, then the future
behavior is predictable,..
BOLZMANN says: yes, just solve my phase space integral
(for static properties)
Molsim 2012
THE question:
“Can we predict the
macroscopic properties of
(classical) many-body systems?”
NEWTON says: yes, my F=ma gives
the future time evolution of the
system
LAPLACE says: in principle yes,
provided that we know the
position, velocity and interaction of
all molecules, then the future
behavior is predictable,..
BOLZMANN says: yes, just solve my phase space integral
(for static properties)
both approaches should lead to the same results:
Molsim 2012
THE question:
“Can we predict the
macroscopic properties of
(classical) many-body systems?”
NEWTON says: yes, my F=ma gives
the future time evolution of the
system
LAPLACE says: in principle yes,
provided that we know the
position, velocity and interaction of
all molecules, then the future
behavior is predictable,..
BOLZMANN says: yes, just solve my phase space integral
(for static properties)
both approaches should lead to the same results:
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But......
.... there are so many molecules....
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.... there are so many molecules....
...making these approaches untractable.
What was the alternative at the time ?
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.... there are so many molecules....
...making these approaches untractable.
What was the alternative at the time ?
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.... there are so many molecules....
...making these approaches untractable.
What was the alternative at the time ?
1. Smart tricks (“theory”)
Only works in special cases: the Ising model: the ideal gas, etc etc
Molsim 2012
.... there are so many molecules....
...making these approaches untractable.
What was the alternative at the time ?
1. Smart tricks (“theory”)
Only works in special cases: the Ising model: the ideal gas, etc etc
2. Constructing a model (“molecular lego”).
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J.D. Bernal’s ball-bearing model
of an atomic liquid
Watson and Crick’s model of
DNA double helix
The computer age
The computer age (1953…)
Berni Alder
Mary-Ann Mansigh
Tom Wainwright
With computers we can follow the behavior of hundreds to
Withhundreds
computers
can follow
the behavior of hundreds to hundreds
of we
millions
of molecules.
of millions of molecules.
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Increment of power since 1950’s
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Super
computer performance
development
Performance
Development
!"%%4'()*+,%
100 Pflop/s
100000000
10 Pflop/s
2#3%4'()*+,%
10000000
1 Pflop/s
1000000
100 Tflop/s
56/%
100000
-0%1'()*+,%
10 Tflop/s
10000
1 Tflop/s
1000
780%
0#0$%1'()*+,%
6-8 years
100 Gflop/s
100
78!..%
!"#$%&'()*+,%
My Laptop (12 Gflop/s)
10 Gflop/s
10
My iPad2 & iPhone 4s (1.02 Gflop/s)
1 Gflop/s
1
100 Mflop/s
0.1
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-..%/'()*+,%
Speed of MD simulations
Psec/atom/CPU h
1012
106
1
1950
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1970
1990
2010
Speed of MD simulations
Psec/atom/CPU h
1012
ab-initio MD is
invented
106
1
1950
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1970
1990
2010
Uses of Molecular Simulations
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Uses of Molecular Simulations
• Mimic the real world:
– Predicting properties of (new) materials
– Computer ‘experiments’ at extreme conditions
– Understanding phenomena on a molecular scale
Molsim 2012
Uses of Molecular Simulations
• Mimic the real world:
– Predicting properties of (new) materials
– Computer ‘experiments’ at extreme conditions
– Understanding phenomena on a molecular scale
• Model systems
– test theory using same simple model
– explore consequences of model
– explain poorly understood phenomena in terms of
essential physics
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Properties of materials
Critical properties of long chain hydrocarbons
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Properties of materials
Critical properties of long chain hydrocarbons
To predict the thermodynamic properties (boiling points)
of the hydrocarbon mixtures it is convenient to know the critical
points of the hydrocarbons.
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Hydrocarbons intermolecular potential
United-atom model
–
–
–
–
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Fixed bond length
Bond-bending
Torsion
Non-bonded: Lennard-Jones
CH2
CH2
CH3
CH2
CH3
Vapour-liquid
equilibria
13
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Vapour-liquid
equilibria
Computational issues:
13
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Vapour-liquid
equilibria
Computational issues:
• How to compute
vapour-liquid
equilibrium?
13
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Vapour-liquid
equilibria
Computational issues:
• How to compute
vapour-liquid
equilibrium?
• How to deal with long
chain hydrocarbons?
13
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Vapour-liquid
equilibria
Course on Free
Energies and Phase
Equilibrium
Computational issues:
• How to compute
vapour-liquid
equilibrium?
• How to deal with long
chain hydrocarbons?
13
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Vapour-liquid
equilibria
Course on Free
Energies and Phase
Equilibrium
Computational issues:
• How to compute
vapour-liquid
equilibrium?
• How to deal with long
chain hydrocarbons?
Course on CBMC:
configurational-bias
Monte Carlo
13
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Properties at extreme conditions
•
•
Carbon phase behavior at very high pressure and temperature
Empirical pair potential depending on carbon coordination
coordination number
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Properties at extreme conditions
•
•
Carbon phase behavior at very high pressure and temperature
Empirical pair potential depending on carbon coordination
coordination number
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Understand molecular processes
•
•
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Protein conformational change with Molecular dynamics
Empirical potential, including bonds, angles dihedrals
Understand molecular processes
•
•
Protein conformational change with Molecular dynamics
Empirical potential, including bonds, angles dihedrals
Course on MD
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Understand molecular processes
Molsim 2012
•
•
Protein conformational change with Molecular dynamics
Empirical potential, including bonds, angles dihedrals
•
Transition path sampling
Understand molecular processes
•
•
Protein conformational change with Molecular dynamics
Empirical potential, including bonds, angles dihedrals
•
Transition path sampling
Course on
rare events
Molsim 2012
Understand molecular processes
Molsim 2012
•
•
Protein conformational change with Molecular dynamics
Empirical potential, including bonds, angles dihedrals
•
Transition path sampling
Understand molecular processes
Molsim 2012
•
•
Protein conformational change with Molecular dynamics
Empirical potential, including bonds, angles dihedrals
•
Transition path sampling
Understand molecular processes
•
•
Protein conformational change with Molecular dynamics
Empirical potential, including bonds, angles dihedrals
•
Transition path sampling
Leads to insight and
new hypotheses
Molsim 2012
Testing theories with simple models
Molsim 2012
Testing theories with simple models
In the 1950‘s an important question arose: Is an attractive interaction
always required to form a solid phase?
Molsim 2012
Testing theories with simple models
In the 1950‘s an important question arose: Is an attractive interaction
always required to form a solid phase?
YES:
– Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids
Molsim 2012
Testing theories with simple models
In the 1950‘s an important question arose: Is an attractive interaction
always required to form a solid phase?
YES:
– Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids
MAYBE NOT:
– These theories do not apply to solids
Molsim 2012
Testing theories with simple models
In the 1950‘s an important question arose: Is an attractive interaction
always required to form a solid phase?
YES:
– Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids
MAYBE NOT:
– These theories do not apply to solids
HOWEVER:
– There no are molecules with only repulsive interactions
Molsim 2012
Testing theories with simple models
In the 1950‘s an important question arose: Is an attractive interaction
always required to form a solid phase?
YES:
– Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids
MAYBE NOT:
– These theories do not apply to solids
HOWEVER:
– There no are molecules with only repulsive interactions
So how to test the hypothesis that molecules with only repulsive
interaction can freeze?
Molsim 2012
Testing theories with simple models
In the 1950‘s an important question arose: Is an attractive interaction
always required to form a solid phase?
YES:
– Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids
MAYBE NOT:
– These theories do not apply to solids
HOWEVER:
– There no are molecules with only repulsive interactions
So how to test the hypothesis that molecules with only repulsive
interaction can freeze?
Molsim 2012
Testing theories with simple models
In the 1950‘s an important question arose: Is an attractive interaction
always required to form a solid phase?
YES:
– Because theories (up to then) predict that attractions are needed for vapourliquid equilibrium and thus why not for solids
MAYBE NOT:
– These theories do not apply to solids
HOWEVER:
– There no are molecules with only repulsive interactions
So how to test the hypothesis that molecules with only repulsive
Your
hypothesis
is
interaction
can freeze?
WRONG it disagrees
with the experiments
Molsim 2012
Testing theories with simple models
My hypothesis is
In the 1950‘s an important question arose: Is an attractive interaction
RIGHT: but this
always required to form a solid phase?
experimentalist refuses
YES:
to usearemolecules
that
– Because theories (up to then) predict that attractions
needed for vapourliquid equilibrium and thus why not for solids
do not have any
MAYBE NOT:
attractive interactions
– These theories do not apply to solids
HOWEVER:
– There no are molecules with only repulsive interactions
So how to test the hypothesis that molecules with only repulsive
Your
hypothesis
is
interaction
can freeze?
WRONG it disagrees
with the experiments
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Simulation of hard spheres
An older Alder
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Simulation of hard spheres
• Bernie Alder et al. carried out
Molecular Dynamics simulations of
the freezing of hard spheres
An older Alder
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Simulation of hard spheres
• Bernie Alder et al. carried out
Molecular Dynamics simulations of
the freezing of hard spheres
• But, …. did the scientific
community accept this computer
results as evidence …?
An older Alder
Molsim 2012
Simulation of hard spheres
• Bernie Alder et al. carried out
Molecular Dynamics simulations of
the freezing of hard spheres
• But, …. did the scientific
community accept this computer
results as evidence …?
• … during a New Jersey conference
in 1957 it was proposed to vote on
it …
An older Alder
Molsim 2012
Simulation of hard spheres
• Bernie Alder et al. carried out
Molecular Dynamics simulations of
the freezing of hard spheres
• But, …. did the scientific
community accept this computer
results as evidence …?
• … during a New Jersey conference
in 1957 it was proposed to vote on
it …
• … and it was voted against the
results of Alder!
An older Alder
Molsim 2012
Experiments are now possible
.. not on molecules but on colloids
from :Yethiraj and van Blaaderen
Nature 421, 513-517 (2003)
and show that hard spheres indeed crystallize at high density
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Explore consequences of model
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•
Trans membrane peptides
•
Coarse-grained model of a membrane to study the interactions
between peptides in a membrane
Explore consequences of model
•
Trans membrane peptides
•
Coarse-grained model of a membrane to study the interactions
between peptides in a membrane
Course on
coarse graining
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Explore consequences of model
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•
Trans membrane peptides
•
Coarse-grained model of a membrane to study the interactions
between peptides in a membrane
Explore consequences of model
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•
Trans membrane peptides
•
Coarse-grained model of a membrane to study the interactions
between peptides in a membrane
Understanding in terms of essential physics
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Understanding in terms of essential physics
•
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Protein folding is complex:
explicit MD intractable
Understanding in terms of essential physics
•
Protein folding is complex:
explicit MD intractable
•
Make simple lattice model with
essential physics of proteins
– polymer connectivity
– heteropolymer sequence
– attraction/repulsion
Molsim 2012
Understanding in terms of essential physics
•
Protein folding is complex:
explicit MD intractable
•
Make simple lattice model with
essential physics of proteins
– polymer connectivity
– heteropolymer sequence
– attraction/repulsion
Molsim 2012
Understanding in terms of essential physics
•
Protein folding is complex:
explicit MD intractable
•
Make simple lattice model with
essential physics of proteins
– polymer connectivity
– heteropolymer sequence
– attraction/repulsion
•
Molsim 2012
MC simulation yields understanding
Understanding in terms of essential physics
•
Protein folding is complex:
explicit MD intractable
•
Make simple lattice model with
essential physics of proteins
– polymer connectivity
– heteropolymer sequence
– attraction/repulsion
•
MC simulation yields understanding
Course on
lattice models
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The limits of Molecular Simulation
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•
Brute-force simulations can never bridge all the scales between
microscopic (nanometers/picoseconds) and macroscopic (cells,
humans, planets).
•
Need different levels of
description (“coarse
graining”) - and we need
input from experiments
at many different levels
to validate our models.
The limits of experiments
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The limits of experiments
•
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Increasingly, experiments generate far more data than humans can digest.
The limits of experiments
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•
Increasingly, experiments generate far more data than humans can digest.
•
Result: “Experulation” or “Simuriment”.
The limits of experiments
Molsim 2012
•
Increasingly, experiments generate far more data than humans can digest.
•
Result: “Experulation” or “Simuriment”.
•
Simulations are becoming an integral part of the analysis of experimental
data.
Understanding Molecular Simulation
Molecular simulations are based on the framework of statistical
mechanics/thermodynamics
Hence:
Molsim 2012
Understanding Molecular Simulation
Molecular simulations are based on the framework of statistical
mechanics/thermodynamics
Hence:
An introduction
(or refresher) of
Statistical
Thermodynamics
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Outline
• Basic Assumption
– micro-canonical ensemble
– relation to thermodynamics
• Canonical ensemble
– free energy
– thermodynamic properties
• Other ensembles
– constant pressure
– grand-canonical ensemble
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Statistical Thermodynamics: the basics
Molsim 2012
Statistical Thermodynamics: the basics
• Nature is quantum-mechanical
Molsim 2012
Statistical Thermodynamics: the basics
• Nature is quantum-mechanical
• Consequence:
– Systems have discrete quantum states.
– For finite “closed” systems, the number of states is finite (but usually
very large)
Molsim 2012
Statistical Thermodynamics: the basics
• Nature is quantum-mechanical
• Consequence:
– Systems have discrete quantum states.
– For finite “closed” systems, the number of states is finite (but usually
very large)
• Hypothesis: In a closed system, every state is equally likely to be
observed.
Molsim 2012
Statistical Thermodynamics: the basics
• Nature is quantum-mechanical
• Consequence:
– Systems have discrete quantum states.
– For finite “closed” systems, the number of states is finite (but usually
very large)
• Hypothesis: In a closed system, every state is equally likely to be
observed.
• Consequence:
ALL of equilibrium Statistical Mechanics and Thermodynamics
Molsim 2012
Simpler example: standard statistics
First: Simpler example (standard statistics)
Draw N balls from an infinite vessel that contains an equal
Draw of
N red
ballsand
from
an balls
infinite vessel that contains an
number
blue
equal number of red and blue balls
!
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!
1.0
0.8
0.6
0.4
0.2
0.2
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0.4
0.6
NR/N
0.8
1.0
1.0
0.8
N=10
0.6
0.4
0.2
0.2
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0.4
0.6
NR/N
0.8
1.0
1.0
0.8
N=10
N=100
0.6
0.4
0.2
0.2
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0.4
0.6
NR/N
0.8
1.0
N=1000
1.0
0.8
N=10
N=100
0.6
0.4
0.2
0.2
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0.4
0.6
NR/N
0.8
1.0
N=1000
N=10000
1.0
0.8
N=10
N=100
0.6
0.4
0.2
0.2
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0.4
0.6
NR/N
0.8
1.0
Molecular consequence of
basic assumption
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Molecular consequence of
basic assumption
Each individual
microstate is equally
probable
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…, but there
are not many
Molecular consequence
of
microstates that give these
basic assumption extreme results
Each individual
microstate is equally
probable
Molsim 2012
…, but there
are not many
Molecular consequence
of
microstates that give these
basic assumption extreme results
Each individual
microstate is equally
probable
If the number of particles
is large (>100) the
probability functions for
these states are sharply
peaked
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Does the basis assumption lead to something that is
consistent with classical thermodynamics?
Molsim 2012
Does the basis assumption lead to something that is
consistent with classical thermodynamics?
Systems 1 and 2 are weakly coupled
such that they can exchange energy.
What will be E1?
Molsim 2012
Does the basis assumption lead to something that is
consistent with classical thermodynamics?
Systems 1 and 2 are weakly coupled
such that they can exchange energy.
What will be E1?
Molsim 2012
Does the basis assumption lead to something that is
consistent with classical thermodynamics?
Systems 1 and 2 are weakly coupled
such that they can exchange energy.
What will be E1?
BA: each configuration is equally probable; but the number of states that
give an energy E1 is not (yet) known.
Molsim 2012
Does the basis assumption lead to something that is
consistent with classical thermodynamics?
Systems 1 and 2 are weakly coupled
such that they can exchange energy.
What will be E1?
BA: each configuration is equally probable; but the number of states that
give an energy E1 is not (yet) known.
The most likely E1, is the E1 that maximizes Ω1(E1)x Ω2(E-E1)
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( )
(
)
⎛ ∂ ln Ω1 E1 ⎞
⎛ ∂ ln Ω2 E − E1 ⎞
+ ⎜
⎜
⎟
⎟
∂E1
∂E1
⎝
⎠ N ,V ⎝
⎠ N
1
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1
=0
2 ,V2
Energy is conserved!
dE1=-dE2
( )
(
)
⎛ ∂ ln Ω1 E1 ⎞
⎛ ∂ ln Ω2 E − E1 ⎞
+ ⎜
⎜
⎟
⎟
∂E1
∂E1
⎝
⎠ N ,V ⎝
⎠ N
1
Molsim 2012
1
=0
2 ,V2
Energy is conserved!
dE1=-dE2
( )
(
)
⎛ ∂ ln Ω1 E1 ⎞
⎛ ∂ ln Ω2 E − E1 ⎞
+ ⎜
⎜
⎟
⎟
∂E1
∂E1
⎝
⎠ N ,V ⎝
⎠ N
1
Molsim 2012
1
=0
2 ,V2
Energy is conserved!
dE1=-dE2
( )
(
)
⎛ ∂ ln Ω1 E1 ⎞
⎛ ∂ ln Ω2 E − E1 ⎞
+ ⎜
⎜
⎟
⎟
∂E1
∂E1
⎝
⎠ N ,V ⎝
⎠ N
1
1
=0
2 ,V2
This can be seen as an
equilibrium condition
Molsim 2012
Energy is conserved!
dE1=-dE2
( )
(
)
⎛ ∂ ln Ω1 E1 ⎞
⎛ ∂ ln Ω2 E − E1 ⎞
+ ⎜
⎜
⎟
⎟
∂E1
∂E1
⎝
⎠ N ,V ⎝
⎠ N
1
1
=0
2 ,V2
This can be seen as an
equilibrium condition
Indeed this is the condition for thermal equilibrium:
“no spontaneous heat flow between 1 and 2”
Molsim 2012
Normally, thermal equilibrium means: equal temperatures
Let us define:
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Normally, thermal equilibrium means: equal temperatures
Let us define:
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Normally, thermal equilibrium means: equal temperatures
Let us define:
Then, thermal equilibrium is equivalent to:
This suggests that β is a function of T.
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Relation to thermodynamics
Conjecture:
Almost right.
Good features:
– Extensivity
– Third law of thermodynamics comes for free
Bad feature:
– It assumes that entropy is dimensionless but (for unfortunate,
historical reasons, it is not…)
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Relation to thermodynamics
Conjecture:
Almost right.
Good features:
– Extensivity
– Third law of thermodynamics comes for free
Bad feature:
– It assumes that entropy is dimensionless but (for unfortunate,
historical reasons, it is not…)
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We have to live with the past, therefore
With kB= 1.380662 10-23 J/K
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We have to live with the past, therefore
With kB= 1.380662 10-23 J/K
In thermodynamics, the absolute (Kelvin) temperature scale
was defined such that
Molsim 2012
We have to live with the past, therefore
With kB= 1.380662 10-23 J/K
In thermodynamics, the absolute (Kelvin) temperature scale
was defined such that
Molsim 2012
We have to live with the past, therefore
With kB= 1.380662 10-23 J/K
In thermodynamics, the absolute (Kelvin) temperature scale
was defined such that
n
dE = TdS-pdV + ∑ µi dN i
i =1
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We have to live with the past, therefore
With kB= 1.380662 10-23 J/K
In thermodynamics, the absolute (Kelvin) temperature scale
was defined such that
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We have to live with the past, therefore
With kB= 1.380662 10-23 J/K
In thermodynamics, the absolute (Kelvin) temperature scale
was defined such that
But we found (defined):
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And this gives the “statistical” definition of temperature:
with
1
β=
kB T
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And this gives the “statistical” definition of temperature:
with
1
β=
kB T
In short:
Entropy and temperature are both related
to the fact that we can COUNT states.
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Number of configurations
35
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Number of configurations
How large is Ω?
35
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Number of configurations
How large is Ω?
For macroscopic systems, super-astronomically large.
35
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Number of configurations
How large is Ω?
For macroscopic systems, super-astronomically large.
For instance, for a glass of water at room temperature:
35
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Number of configurations
How large is Ω?
For macroscopic systems, super-astronomically large.
For instance, for a glass of water at room temperature:
35
Molsim 2012
Number of configurations
How large is Ω?
For macroscopic systems, super-astronomically large.
For instance, for a glass of water at room temperature:
Macroscopic deviations from the second law of thermodynamics are not
forbidden, but they are extremely unlikely.
35
Molsim 2012
Deviation from the 2nd law?
What is the probability that 1 mole of argon gas spontaneously lowers it
entropy by 0.00000001% (=10-10) ?
standard molar entropy of argon : Sargon =307.2 J/K mol-1
slightly lower entropy Sargon,low =307.2 (1 - 10-10) J/K mol-1
probability of occurrence is
Molsim 2012
Deviation from the 2nd law?
What is the probability that 1 mole of argon gas spontaneously lowers it
entropy by 0.00000001% (=10-10) ?
standard molar entropy of argon : Sargon =307.2 J/K mol-1
slightly lower entropy Sargon,low =307.2 (1 - 10-10) J/K mol-1
probability of occurrence is
�
�
�
−8
Ωargon,low
Sargon,low − Sargon
−3.07 × 10
P ≈
= exp
= exp
Ωargon
kB
1.38 × 10−23
Molsim 2012
�
−1015
≈ 10
Deviation from the 2nd law?
What is the probability that 1 mole of argon gas spontaneously lowers it
entropy by 0.00000001% (=10-10) ?
standard molar entropy of argon : Sargon =307.2 J/K mol-1
slightly lower entropy Sargon,low =307.2 (1 - 10-10) J/K mol-1
probability of occurrence is
�
�
�
−8
Ωargon,low
Sargon,low − Sargon
−3.07 × 10
P ≈
= exp
= exp
Ωargon
kB
1.38 × 10−23
A mathematical relation : 10-1015 ≠ 0
A physical relation : 10-1015 = 0
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�
−1015
≈ 10
Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
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Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
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Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
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1/kBT
Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
Molsim 2012
Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
Molsim 2012
Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
Molsim 2012
Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
Molsim 2012
Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
Partition�
function:
Q=
exp(−Ej /kB T )
j
Molsim 2012
Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
Molsim 2012
Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
Molsim 2012
Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
Boltzmann distribution
Molsim 2012
Canonical ensemble
Consider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
“Low energies are more likely than high energies”
Molsim 2012
Thermodynamics
What is the average energy of the system?
Molsim 2012
Thermodynamics
What is the average energy of the system?
Molsim 2012
Thermodynamics
What is the average energy of the system?
Molsim 2012
Thermodynamics
What is the average energy of the system?
Molsim 2012
Thermodynamics
What is the average energy of the system?
Compare:
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Thermodynamics
Thermo recall
What is the average
energy ofequation
the system?
Fundamental
Helmholtz Free energy:
Compare:
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Thermodynamics
What is the average energy of the system?
Compare:
Hence:
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The classical limit
We have assume quantum mechanics (discrete states) but
we are interested in the classical limit
N
Particles are indistinguishable
N
“volume of phase space”
Why does planck constant appear?
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Appearance of Plank’s constant?
Easiest to look at translation partition function Qx of particle in 1D box
1 2 p2
"trans = mv =
2
2m
!
Molsim 2012
p= momentum
m=mass
Postulate quantum mechanics: p=h/λ met h Planck’s constant
Particle is standing wave with length
Appearance of Plank’s constant?
Easiest to look at translation partition function Qx of particle in 1D box
p= momentum
m=mass
1 2 p2
"trans = mv =
2
2m
!
Postulate quantum mechanics: p=h/λ met h Planck’s constant
Particle is standing wave with length
"=
2L
n
!
!
Molsim 2012
2
p 2 ( h / #)
n 2h 2
"trans =
=
=
2m
2m
8mL2
Appearance of Plank’s constant?
Easiest to look at translation partition function Qx of particle in 1D box
p= momentum
m=mass
1 2 p2
"trans = mv =
2
2m
!
Postulate quantum mechanics: p=h/λ met h Planck’s constant
Particle is standing wave with length
"=
2L
n
2
p 2 ( h / #)
n 2h 2
"trans =
=
=
2m
2m
8mL2
$ ! n2h2 '
Qx = # exp(!!"trans ) = # exp & !
2 )
8mL
%
(
n=1
n=1
"
!
!
Molsim 2012
"
Appearance of Plank’s constant?
Easiest to look at translation partition function Qx of particle in 1D box
p= momentum
m=mass
1 2 p2
"trans = mv =
2
2m
!
Postulate quantum mechanics: p=h/λ met h Planck’s constant
Particle is standing wave with length
"=
2
p 2 ( h / #)
n 2h 2
"trans =
=
=
2m
2m
8mL2
2L
n
$ ! n2h2 '
Qx = # exp(!!"trans ) = # exp & !
2 )
8mL
%
(
n=1
n=1
"
"
!
!
Qx =
Molsim 2012
#
"
1
dne
! ! n 2 h 2 /(8mL2 )
=
L 2" m
h
!
Appearance of Plank’s constant?
Easiest to look at translation partition function Qx of particle in 1D box
p= momentum
m=mass
1 2 p2
"trans = mv =
2
2m
!
Postulate quantum mechanics: p=h/λ met h Planck’s constant
Particle is standing wave with length
"=
2
p 2 ( h / #)
n 2h 2
"trans =
=
=
2m
2m
8mL2
2L
n
$ ! n2h2 '
Qx = # exp(!!"trans ) = # exp & !
2 )
8mL
%
(
n=1
n=1
"
"
!
!
Qx =
#
"
1
dne
! ! n 2 h 2 /(8mL2 )
=
L 2" m
h
!
now compare
� � to classical integration without Planck’s constant
�
2
Qx =
p
dpdr exp(−β
+ U (r)) = L
2m
factor of h is missing for each degree of freedom.
Molsim 2012
2
p
dp exp(−β
)=L
2m
�
2πm
β
The classical limit
We have assume quantum mechanics (discrete states) but
we are interested in the classical limit
N
Particles are indistinguishable
N
Molsim 2012
Volume of phase space (particle in a box)
The classical limit
We have assume quantum mechanics (discrete states) but
we are interested in the classical limit
N
Particles are indistinguishable
N
Volume of phase space (particle in a box)
Integration over the momenta can be carried out for most systems:
Molsim 2012
The classical limit
Molsim 2012
The classical limit
Define de Broglie wavelength:
Molsim 2012
The classical limit
Define de Broglie wavelength:
Partition function:
Molsim 2012
Example: ideal gas
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Example: ideal gas
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Example: ideal gas
Molsim 2012
Example: ideal gas
Free energy:
! N ln " 3 # N lnV + N ln N # N
Molsim 2012
Example: ideal gas
Free energy:
! N ln " 3 # N lnV + N ln N # N
! F = N ( ln( !! 3 ) "1)
Molsim 2012
with density ρ=N/V
Example: ideal gas
Thermo recall
Helmholtz Free energy:
Free energy: Pressure
! N ln " 3 # N lnV + N ln N # N
3
! F = NEnergy:
ln(
!
!
) "1)
(
Molsim 2012
with density ρ=N/V
Example: ideal gas
Free energy:
! N ln " 3 # N lnV + N ln N # N
! F = N ( ln( !! 3 ) "1)
Molsim 2012
with density ρ=N/V
Example: ideal gas
Free energy:
! N ln " 3 # N lnV + N ln N # N
! F = N ( ln( !! 3 ) "1)
Pressure:
Molsim 2012
with density ρ=N/V
Example: ideal gas
Free energy:
! N ln " 3 # N lnV + N ln N # N
! F = N ( ln( !! 3 ) "1)
Pressure:
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with density ρ=N/V
Energy:
Ideal gas (2)
Chemical potential:
Molsim 2012
⎛ ∂F ⎞
µi = ⎜
⎟
∂N
⎝ i ⎠ T ,V , N j
Ideal gas (2)
⎛ ∂F ⎞
µi = ⎜
⎟
∂N
⎝ i ⎠ T ,V , N j
Chemical potential:
! F = N ( ln( !! ) "1)
3
Molsim 2012
with density ρ=N/V
Ideal gas (2)
⎛ ∂F ⎞
µi = ⎜
⎟
∂N
⎝ i ⎠ T ,V , N j
Chemical potential:
! F = N ( ln( !! ) "1)
3
!µ = ln ! 3 + ln "
Molsim 2012
with density ρ=N/V
Ideal gas (2)
⎛ ∂F ⎞
µi = ⎜
⎟
∂N
⎝ i ⎠ T ,V , N j
Chemical potential:
! F = N ( ln( !! ) "1)
3
!µ = ln ! 3 + ln "
βµ
Molsim 2012
IG
0
= βµ + ln ρ
with density ρ=N/V
Heat capacity from energy fluctuation
! ! E $ ! !" $
!! E $
CV = # & = # & # &
" ! T %V ,N " !" %V ,N " ! T %
Molsim 2012
Heat capacity from energy fluctuation
! ! E $ ! !" $
!! E $
CV = # & = # & # &
" ! T %V ,N " !" %V ,N " ! T %
Molsim 2012
"! E %
kBT CV = ! $ '
# !" &V ,N
2
Heat capacity from energy fluctuation
! ! E $ ! !" $
!! E $
CV = # & = # & # &
" ! T %V ,N " !" %V ,N " ! T %
"! E %
kBT CV = ! $ '
# !" &V ,N
2
" E exp(!! E )
=
" exp(!! E )
i
E
i
i
i
i
Molsim 2012
Heat capacity from energy fluctuation
"! E %
kBT CV = ! $ '
# !" &V ,N
! ! E $ ! !" $
!! E $
CV = # & = # & # &
" ! T %V ,N " !" %V ,N " ! T %
E exp(!" E )
"
!
=!
!" " exp(!" E )
i
kBT 2CV
2
i
i
i
i
" E exp(!" Ei ) #% " Ei exp(!" Ei ) &(
= i
!% i
(
exp(!
"
E
)
exp(!
"
E
)
"
i
i
% "
(
$ i
'
i
2
i
= E
Molsim 2012
2
! E
2
=
(E !
E
)
2
2
Heat capacity from energy fluctuation
"! E %
kBT CV = ! $ '
# !" &V ,N
! ! E $ ! !" $
!! E $
CV = # & = # & # &
" ! T %V ,N " !" %V ,N " ! T %
E exp(!" E )
"
!
=!
!" " exp(!" E )
i
kBT 2CV
2
i
i
i
i
" E exp(!" Ei ) #% " Ei exp(!" Ei ) &(
= i
!% i
(
exp(!
"
E
)
exp(!
"
E
)
"
i
i
% "
(
$ i
'
i
2
i
= E
Molsim 2012
2
! E
2
=
(E !
E
)
2
P(E)
fluctuation in E
grows as 1/√N
2
!"
Computing the pressure
∂F
P =−
∂V
F = −kT ln Q
Introduce “scaled” coordinates:
Introduce scaled coordinates
Molsim 2012
Computing the pressure
Molsim 2012
Molsim 2012
Free energy derivative can be written as ensemble average!
Molsim 2012
For pairwise additive forces:
Then
Molsim 2012
i and j are dummy variable hence:
And we can write
Molsim 2012
But as action equals reaction (Newton’s 3rd law):
And hence
Inserting this in our expression for the pressure, we get:
Where
This is known as the virial expression
Molsim 2012
Whatdo
to do
youif cannot
useuse
the the
virial
expression?
What
youif do
you can’t
virial
expression?
This will be the case e.g. with discontinuous potentials.
Molsim 2012
Summary:
micro-canonical ensemble (N,V,E)
Molsim 2012
Summary:
micro-canonical ensemble (N,V,E)
Partition function:
Molsim 2012
Summary:
micro-canonical ensemble (N,V,E)
Partition function:
Probability to find a particular configuration
Molsim 2012
Summary:
micro-canonical ensemble (N,V,E)
Partition function:
Probability to find a particular configuration
Entropy
S = kB lnQN,V ,E
Molsim 2012
Summary:
Canonical ensemble (N,V,T)
Molsim 2012
Summary:
Canonical ensemble (N,V,T)
Partition function:
Molsim 2012
Summary:
Canonical ensemble (N,V,T)
Partition function:
Probability to find a particular configuration
Molsim 2012
Summary:
Canonical ensemble (N,V,T)
Partition function:
Probability to find a particular configuration
Free energy
Molsim 2012
Ensemble averages
For properties that only depend on the configurational part
Molsim 2012
Ensemble averages
For properties that only depend on the configurational part
Probability to find a configuration:
Molsim 2012
Ensemble averages
For properties that only depend on the configurational part
Probability to find a configuration:
1
P(!) = exp ["!U(!)]
Q
Molsim 2012
Ensemble averages
For properties that only depend on the configurational part
Probability to find a configuration:
1
P(!) = exp ["!U(!)]
Q
Ensemble average:
A =
Molsim 2012
" d!A(!)P(!)
Ergodicity theorem
suppose we have an ensemble average of a system defined by U(Γ) obtained by MC
Now suppose we have a NVT molecular dynamics trajectory for the same system
A time average over the trajectory is simply
Molsim 2012
Ergodicity theorem
suppose we have an ensemble average of a system defined by U(Γ) obtained by MC
Now suppose we have a NVT molecular dynamics trajectory for the same system
A time average over the trajectory is simply
Ergodicity theorem states that for an ‘ergodic system’
Molsim 2012
Ergodicity theorem
suppose we have an ensemble average of a system defined by U(Γ) obtained by MC
Now suppose we have a NVT molecular dynamics trajectory for the same system
A time average over the trajectory is simply
Ergodicity theorem states that for an ‘ergodic system’
MC and MD give the same averages
Molsim 2012
Other ensembles?
Molsim 2012
Other ensembles?
In the thermodynamic limit the thermodynamic properties are
independent of the ensemble: so buy a bigger computer …
Molsim 2012
Other ensembles?
In the thermodynamic limit the thermodynamic properties are
independent of the ensemble: so buy a bigger computer …
However, it is most of the times much better to think and to carefully
select an appropriate ensemble.
Molsim 2012
Other ensembles?
In the thermodynamic limit the thermodynamic properties are
independent of the ensemble: so buy a bigger computer …
However, it is most of the times much better to think and to carefully
select an appropriate ensemble.
For this it is important to know how to simulate in the various ensembles.
Molsim 2012
Other ensembles?
In the thermodynamic limit the thermodynamic
properties
are
Course
on
independent of the ensemble: so buy a bigger
MDcomputer
and MC in…
different ensembles
However, it is most of the times much better to think and to carefully
select an appropriate ensemble.
For this it is important to know how to simulate in the various ensembles.
Molsim 2012
Other ensembles?
In the thermodynamic limit the thermodynamic properties are
independent of the ensemble: so buy a bigger computer …
However, it is most of the times much better to think and to carefully
select an appropriate ensemble.
For this it is important to know how to simulate in the various ensembles.
Molsim 2012
Other ensembles?
In the thermodynamic limit the thermodynamic properties are
independent of the ensemble: so buy a bigger computer …
However, it is most of the times much better to think and to carefully
select an appropriate ensemble.
For this it is important to know how to simulate in the various ensembles.
But for doing this we need to know the Statistical Thermodynamics
of the various ensembles.
Molsim 2012
Constant pressure simulations:
N,P,T ensemble
Consider a small system that can exchange volume
and energy with a big reservoir
Molsim 2012
Constant pressure simulations:
N,P,T ensemble
Consider a small system that can exchange volume
and energy with a big reservoir
Molsim 2012
Constant pressure simulations:
N,P,T ensemble
1/kBT
p/kBT
Consider a small system that can exchange volume
and energy with a big reservoir
Molsim 2012
Constant pressure simulations:
N,P,T ensemble
Thermo recall
1/kBT
p/kBT
Fundamental
equation
Consider
a small system that can exchange volume
and energy with a big reservoir
Hence
and
Molsim 2012
Constant pressure simulations:
N,P,T ensemble
1/kBT
p/kBT
Consider a small system that can exchange volume
and energy with a big reservoir
Molsim 2012
Constant pressure simulations:
N,P,T ensemble
Consider a small system that can exchange volume
and energy with a big reservoir
Molsim 2012
Constant pressure simulations:
N,P,T ensemble
Consider a small system that can exchange volume
and energy with a big reservoir
Molsim 2012
Constant pressure simulations:
N,P,T ensemble
Consider a small system that can exchange volume
and energy with a big reservoir
Hence, the probability to find Ei,Vi:
Molsim 2012
Constant pressure simulations:
N,P,T ensemble
Consider a small system that can exchange volume
and energy with a big reservoir
Hence, the probability to find Ei,Vi:
Molsim 2012
N,P,T ensemble (2)
In the classical limit, the partition function becomes
Molsim 2012
N,P,T ensemble (2)
In the classical limit, the partition function becomes
Molsim 2012
N,P,T ensemble (2)
In the classical limit, the partition function becomes
The probability to find a particular configuration:
Molsim 2012
N,P,T ensemble (2)
In the classical limit, the partition function becomes
The probability to find a particular configuration:
Molsim 2012
Grand-canonical simulations:
µ,V,T ensemble
Consider a small system that can exchange particles
and energy with a big reservoir
Molsim 2012
Grand-canonical simulations:
µ,V,T ensemble
Consider a small system that can exchange particles
and energy with a big reservoir
Molsim 2012
Grand-canonical simulations:
µ,V,T ensemble
1/kBT
Consider a small system that can exchange particles
and energy with a big reservoir
Molsim 2012
Grand-canonical simulations:
µ,V,T ensemble
-µ/kBT
1/kBT
Consider a small system that can exchange particles
and energy with a big reservoir
Molsim 2012
Grand-canonical simulations:
µ,V,T ensemble
Thermo recall
-µ/kBT
1/kBT
Fundamental
equation
Consider
a small system that can exchange particles
and energy with a big reservoir
Hence
and
Molsim 2012
Grand-canonical simulations:
µ,V,T ensemble
-µ/kBT
1/kBT
Consider a small system that can exchange particles
and energy with a big reservoir
Molsim 2012
Grand-canonical simulations:
µ,V,T ensemble
Consider a small system that can exchange particles
and energy with a big reservoir
Molsim 2012
Grand-canonical simulations:
µ,V,T ensemble
Consider a small system that can exchange particles
and energy with a big reservoir
Molsim 2012
Grand-canonical simulations:
µ,V,T ensemble
Consider a small system that can exchange particles
and energy with a big reservoir
Hence, the probability to find Ei,Ni:
Molsim 2012
Grand-canonical simulations:
µ,V,T ensemble
Consider a small system that can exchange particles
and energy with a big reservoir
Hence, the probability to find Ei,Ni:
Molsim 2012
µ,V,T ensemble (2)
In the classical limit, the partition function becomes
Molsim 2012
µ,V,T ensemble (2)
In the classical limit, the partition function becomes
Molsim 2012
µ,V,T ensemble (2)
In the classical limit, the partition function becomes
The probability to find a particular configuration:
Molsim 2012
Summary
Molsim 2012
Summary
•
Molsim 2012
Molecular simulation is firmly rooted in equilibrium statistical
mechanics/thermodynamics
Summary
Molsim 2012
•
Molecular simulation is firmly rooted in equilibrium statistical
mechanics/thermodynamics
•
Basis of statistical thermodynamics: configurations with same energy
are equally likely: microcanonical ensemble
Summary
Molsim 2012
•
Molecular simulation is firmly rooted in equilibrium statistical
mechanics/thermodynamics
•
Basis of statistical thermodynamics: configurations with same energy
are equally likely: microcanonical ensemble
•
At constant temperature Boltzmann distribution follows:
canonical ensemble
Summary
Molsim 2012
•
Molecular simulation is firmly rooted in equilibrium statistical
mechanics/thermodynamics
•
Basis of statistical thermodynamics: configurations with same energy
are equally likely: microcanonical ensemble
•
At constant temperature Boltzmann distribution follows:
canonical ensemble
•
other ensembles are isobaric and grand canonical ensembles
Summary
Molsim 2012
•
Molecular simulation is firmly rooted in equilibrium statistical
mechanics/thermodynamics
•
Basis of statistical thermodynamics: configurations with same energy
are equally likely: microcanonical ensemble
•
At constant temperature Boltzmann distribution follows:
canonical ensemble
•
other ensembles are isobaric and grand canonical ensembles
•
MD and MC are two roads to the same equilibrium answer: ergodicity
Summary
Molsim 2012
•
Molecular simulation is firmly rooted in equilibrium statistical
mechanics/thermodynamics
•
Basis of statistical thermodynamics: configurations with same energy
are equally likely: microcanonical ensemble
•
At constant temperature Boltzmann distribution follows:
canonical ensemble
•
other ensembles are isobaric and grand canonical ensembles
•
MD and MC are two roads to the same equilibrium answer: ergodicity
•
MD also gives dynamical properties (viscosity, diffusion etc)
Summary
Molsim 2012
•
Molecular simulation is firmly rooted in equilibrium statistical
mechanics/thermodynamics
•
Basis of statistical thermodynamics: configurations with same energy
are equally likely: microcanonical ensemble
•
At constant temperature Boltzmann distribution follows:
canonical ensemble
•
other ensembles are isobaric and grand canonical ensembles
•
MD and MC are two roads to the same equilibrium answer: ergodicity
•
MD also gives dynamical properties (viscosity, diffusion etc)
•
Rest of the week: MC and MD in depth.
The end
Molsim 2012