An atom, a continuum and a transition quandary
This piece is a reconciliation to my recent exposure to the concept of atomistics, molecular dynamics, and a transition to continuum over several scales.
Atomistics for me, starts with understanding motion of atoms i.e., their nuclei and electrons. Owing to wave particle duality established by Erwin Schrodinger, one could solve for the wave equation and the associated energy of each nucleus, and electrons independently for a given IVP or BVP. However, it is understood that electron motion can be decoupled from motion of nucleus which is several orders of magnitude slower than the former. This postulation attributed to 'Born-Oppenheimer' aides one to account for nucleus motion ahead of electron motion. The latter then can then be established based on pre-computed former's location. Simply put, "know where nuclei are first, for electrons form fast moving clouds around them".
Idea of density function theory (DFT):
Solving for wave equation and energy for each electron is a) computationally expensive as atoms get larger b) adds lesser value for greater effort since, electrons form a 'cloud' around the nucleus. Hence, one could deem the cloud as a effective localized presence of electrons in an atomic disposition, or simply put a region of maximum electron density. One could identify such a region, appealing to first principles of minimization of energy of electron cloud, given nucleus positions. Since, cloud is effectively, area of finding any and all electrons, it is a squared sum of all electron wave functions over the whole domain.
Hamiltonian description of atoms:
When one solves aforementioned wave equation for atomic disposition, one obtains energy and wave form of atomic motion. However, given the atomic mass, one could assume that the wave nature demonstrated by atoms could be much smaller than that of electrons, more so at low temperatures. This leads to a particle type treatment of atoms or simply put 'Hamiltonian description'. In particular, one assumes that the wavelength of atomic motion (de Broglie wavelength) is much smaller than inter.atomic separation so that wave-wave interactions can be neglected.
Here, the idea is to track particles with their position, and derivative of their position or momentum (q, p). For this purpose energy potentials (pair, three body, multi body, ionic etc.) are defined. These potentials are largely empirically obtained or from intuition. These potentials are then used to define forces in atomic ensemble from whence motion of atoms can be tracked.
One such potential is embedded atom method (EAM), which accounts for interaction between atom of interest and all other atoms besides interaction of the former from the electron cloud in it's vicinity.
Note: One of the challenges with this method is, identifying a reasonable potential. For instance, potential that incorporates both long range (Coloumbic interactions), and relatively short range forces. This is of particular interest in Coarse-graining techniques (Non-local QC), discussed below.
Idea of Phase Space and molecular dynamics:
An atom can be traced if its location and direction of motion are known i.e., position and velocity/momentum (q, p). When one solves aforementioned wave equation for atomic disposition, one obtains {(q, p)} of all atoms. This is computationally expensive for atomic size is in Angstroms and hence it limits the size over which such computations can be performed (a few hundred nm).
Note: Here {(q, p)} refers to set of all positions and momenta of all atoms in a system.
In particular, when one is interested in macro-scale properties of a system with a large number of atoms, it becomes prohibitively expensive to track the position, and momentum of all atoms individually for one is interested in macroscopic properties that are a 'manifestation of an average over from atomic scale motion'. For instance, for a given macroscopic scale property A({(q, p)}), many choices of {(q, p)} for all atoms are possible. So, for N atoms {(q, p)} represents a 2N dimensional space, where A can be treated as an average of say n (many) choices of {(q, p)}.
Alternatively, one could define a surface such that, all such choices of {(q, p)} lie on it. One could view this as follows: Say, if one wants to track {(q, p)} of a rigid triangle with atoms at the corners, the choice of locations of corners can be reduced from R3xR3XR3 to a sample space Γ by imposing distance between them as a constraint. Similarly, an 'imposed' A, reduces 2N dimensional space to a much smaller domain, which could be visualized in 2D, if {(q, p)} is used.
Essentially, one is interested in the cloud of miscroscopic states{(q, p)} that satisfies a given macrostate A. This cloud is defined as a 'distribution function' or 'density function'.
Q) Why does one need such a 'cloud'?
A: It provides a computationally convenient form to consider those compliant microstates given A, to evaluate say energy/temperature etc. of the system. Obtaining such a distribution function depends on the macrostate A, imposed. Microcanonical ensemble/canonical etc.
to be continued.....
Atomistics for me, starts with understanding motion of atoms i.e., their nuclei and electrons. Owing to wave particle duality established by Erwin Schrodinger, one could solve for the wave equation and the associated energy of each nucleus, and electrons independently for a given IVP or BVP. However, it is understood that electron motion can be decoupled from motion of nucleus which is several orders of magnitude slower than the former. This postulation attributed to 'Born-Oppenheimer' aides one to account for nucleus motion ahead of electron motion. The latter then can then be established based on pre-computed former's location. Simply put, "know where nuclei are first, for electrons form fast moving clouds around them".
Idea of density function theory (DFT):
Solving for wave equation and energy for each electron is a) computationally expensive as atoms get larger b) adds lesser value for greater effort since, electrons form a 'cloud' around the nucleus. Hence, one could deem the cloud as a effective localized presence of electrons in an atomic disposition, or simply put a region of maximum electron density. One could identify such a region, appealing to first principles of minimization of energy of electron cloud, given nucleus positions. Since, cloud is effectively, area of finding any and all electrons, it is a squared sum of all electron wave functions over the whole domain.
Hamiltonian description of atoms:
When one solves aforementioned wave equation for atomic disposition, one obtains energy and wave form of atomic motion. However, given the atomic mass, one could assume that the wave nature demonstrated by atoms could be much smaller than that of electrons, more so at low temperatures. This leads to a particle type treatment of atoms or simply put 'Hamiltonian description'. In particular, one assumes that the wavelength of atomic motion (de Broglie wavelength) is much smaller than inter.atomic separation so that wave-wave interactions can be neglected.
Here, the idea is to track particles with their position, and derivative of their position or momentum (q, p). For this purpose energy potentials (pair, three body, multi body, ionic etc.) are defined. These potentials are largely empirically obtained or from intuition. These potentials are then used to define forces in atomic ensemble from whence motion of atoms can be tracked.
One such potential is embedded atom method (EAM), which accounts for interaction between atom of interest and all other atoms besides interaction of the former from the electron cloud in it's vicinity.
Note: One of the challenges with this method is, identifying a reasonable potential. For instance, potential that incorporates both long range (Coloumbic interactions), and relatively short range forces. This is of particular interest in Coarse-graining techniques (Non-local QC), discussed below.
Idea of Phase Space and molecular dynamics:
An atom can be traced if its location and direction of motion are known i.e., position and velocity/momentum (q, p). When one solves aforementioned wave equation for atomic disposition, one obtains {(q, p)} of all atoms. This is computationally expensive for atomic size is in Angstroms and hence it limits the size over which such computations can be performed (a few hundred nm).
Note: Here {(q, p)} refers to set of all positions and momenta of all atoms in a system.
In particular, when one is interested in macro-scale properties of a system with a large number of atoms, it becomes prohibitively expensive to track the position, and momentum of all atoms individually for one is interested in macroscopic properties that are a 'manifestation of an average over from atomic scale motion'. For instance, for a given macroscopic scale property A({(q, p)}), many choices of {(q, p)} for all atoms are possible. So, for N atoms {(q, p)} represents a 2N dimensional space, where A can be treated as an average of say n (many) choices of {(q, p)}.
Alternatively, one could define a surface such that, all such choices of {(q, p)} lie on it. One could view this as follows: Say, if one wants to track {(q, p)} of a rigid triangle with atoms at the corners, the choice of locations of corners can be reduced from R3xR3XR3 to a sample space Γ by imposing distance between them as a constraint. Similarly, an 'imposed' A, reduces 2N dimensional space to a much smaller domain, which could be visualized in 2D, if {(q, p)} is used.
Essentially, one is interested in the cloud of miscroscopic states{(q, p)} that satisfies a given macrostate A. This cloud is defined as a 'distribution function' or 'density function'.
Q) Why does one need such a 'cloud'?
A: It provides a computationally convenient form to consider those compliant microstates given A, to evaluate say energy/temperature etc. of the system. Obtaining such a distribution function depends on the macrostate A, imposed. Microcanonical ensemble/canonical etc.
to be continued.....
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