Term
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Definition
| Use as few degrees of freedom as possible and certainly many fewer tan the total number of data points |
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Term
| reduced X2(chi squared statistic) |
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Definition
| (1/D.O.F)summation (residuals squared over standard deviation squared |
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Term
| What do different values of chi squared indicate |
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Definition
| 1- good fit, >>1 bad fit, <1 over-fitted |
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Term
| maximum likelihood estimation |
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Definition
| a parameter that estimation method that can be used to limit the model complexity to a level fundamentally compatible with data and their inherent precision. L(theta) = p(xi, x2,..) where p is a probability density function with parameters theta |
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Term
| Structure of simple Artificial Neural Networks |
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Definition
| input data are weighted and feeded into a hidden layer. In the hidden layer the weighted inputs are put into a transfer function and then added together. tanh(x) or exponentials are common transfer functions. In a linear network,weighted inputs are just summed together with a bias of theta. |
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Term
| What's the standard method for training a neural network? |
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Definition
| using the back propagation algorithm which seeks to minimize a sum of the squares of the residuals. |
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Term
| If an ANN has n hidden units and m input units how many variables does it contain |
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Definition
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Term
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Definition
| an experiment in which pi was estimated by randomly tossing a needle of length l at lines of equal spacing d and counting the number of times the needle intersects the lines. The probability of an intersection (281)/(pi*d) |
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Term
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Definition
| a process where it is possible to reach any state from any other state |
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Term
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Definition
1) start with a system in an arbitrarily chose state: A- evaluate it's energy EA
2) Generate a new state B using an ergodic process
3) if EB< EA, accept state B, else if EB> EA accept state B with a probability given by an arrhenius exponential form
4) keep repeating steps 2 and 3 until the system equilibrates |
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Term
| How does the Ising model work? How is the spin interaction between neighbors modeled? |
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Definition
The ising model is a simple model of a magnet which consists of a number of 2 state(up/down) spins on a lattice. Each spin can interact with its nearest neighbour, and also with an external magnetic field:
H= -epsilon summation(SiSj, i and j) - B summation(Si, j) where Si is the spin of the state and Sj is the spin state of neighbors |
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Term
| mean magnetism and heat capacity given by the 2D ising model |
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Definition
= (1/N) summation(Si, i)
c = (k beta^2/N)(- ^2) |
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Term
| What is the price paid for adding unphysical moves to the Metropolis algorhithm? What limitations are set on the kinds of moves you can add? |
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Definition
| The moves must be thermodynamicaly stable and ergodic atleast in principal. By adding unphysical moves you lose information about the dynamics of the system. |
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Term
| What are kinetic Monte Carlo Methods |
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Definition
| Monte Carlo Methods that use reasonably physical move sets and can therefore be used to yield valuable information about the equilibrium dynamics of the system, despite the fact that Monte Carlo processes occur across probability space rather than time. |
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Term
| What do you expect grain growth to follow according to classical theory? How does Potts model work to give a more accurate picture of grain growth? |
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Definition
D_ = K t^n; where D_ is the average grain size, t is the time , and n is usually 1/2. Measurements indicate n can vary between .2 and .5
Pott's model:
a fixed lattice is defined- each site is given a number si- grain boundaries are defines as lines separating different si. The interaction between the neighbors is given by:
H = -J summation(dsisj -1) where dsisj = 1 is si = sj or 0 otherwise. The system therefore prefers states to be oriented in the same direction. If there are only 2 states you have an ising model.
the model then uses the metropolis algorithm to evolve. |
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Term
| How do you model abnormal grain growth? What does classical theory predict for the mean grain size? |
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Definition
Same as normal grain growth just adjust the energy:
H = - J0 summation(dSQk ) - J summation(dsisj -1)
where dSQk = 1 if the grain has the preferred grain alignment
dSiSj= 1 if the two neighboring states have the same state(same as in normal grain growth)
Classical theory predicts that
D_ = (pi*r)/ 2f where r is the particle size of pinning particles, f is the fraction of the material that these particles make up. |
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Term
| In liquid crystals, what is the director? |
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Definition
| The director represents the orientation of the liquid crystal |
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Term
| What is the isotropic-nematic transition? What are other ordered liquid crystal phases? |
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Definition
| When liquid crystal rods change from being randomly aligned to having a net orientational order about a bulk director, n. Other ordered phases are the cholesteric and smectic phase. |
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Term
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Definition
| phases that liquid crystalline polymers show that are very similar to those seen in liquid crystals |
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Term
| What are the the three main classes of elastic distortion that liquid crystalline polymers show? what are the energies associated with each |
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Definition
splay, twist, and bend:
Espaly prop (div n)^2
Etwist prop (n curl n)^2
Eben prop (n X curl n)^2 |
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Term
| Frank free energy density |
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Definition
distortion energies are comined in this to form the energy density of the director field
F = 1/2 [ k11(divn)^2 + k2(ncurln)^2 + k33 (n x curl n)^2]
The magnitudes of the elastic constants are usually k11>k33>k22 |
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Term
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Definition
| point singularities in the director fields observed in the schlieren pattern- analogous to dislocations in crystals. |
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Term
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Definition
| the number of rotations undergone by the director around a closed loop centered on the defect core |
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Term
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Definition
| given by the relative amounts of splay, bend and twist distortions which it contains |
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Term
| What changes the character of disclinations? Why? |
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Definition
| Strong elastic anisotropy i.e in LCPs. This elastic anisotropy causes the material to spontaneously mimize its starin energy. |
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Term
| Ericksen-Leslie equations |
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Definition
| equations governing the motion of LCPs in external fields |
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Term
| What are the most significant effects of liquid crystalline polymer flow? |
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Definition
| liquid crystallinity gives rise to lower viscosity polymers, especially at high strain rates. You get very efficient molecular orientation in extensional strain fields. |
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Term
| What ensembles do MD and MC samples follow? |
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Definition
| MD samples naturally from states at constant energy (microcanonical or NVE ensemble), whereas Metropolis MC samples from states at a constant temperature (canonical or NVT) |
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Term
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Definition
a popular integratir used in MD where the new positions of particals are calculated using positions and accelerations at current time step and old positions:
r(t+deltat) = r(t)+ v(t) deltat + 1/2 a(t)deltat^2 + O(t^3)
r( t- deltat) = r(t)- v(t)deltat + 1/2a(t) deltat^2 - O(t^3)
advantage: high precision, low drift |
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Term
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Definition
| time reversible ( term used in conjunction with MD integrators) |
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Term
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Definition
| time reversible ( term used in conjunction with MD integrators) |
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Term
| What are problems with non-symplectic integrators? |
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Definition
| long-term energy conversion |
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Term
| How do you find the diffusion coefficient using MD simulations? |
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Definition
use the mean-squared discplacement and the Einstein relation:
D = lim(t-> infinity) {|r(t)-r(0)|^2/6t} |
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Term
| How can you adjust MD simulations to get NVT ensemble calculations? |
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Definition
velocity scaling:
= 3/2 N kB T
use calculation of sum of kinetic energy of particles to scale their velocities |
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Term
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Definition
| couple systems to external heat bath |
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Term
| velocity autocorrelation function |
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Definition
| provides a means of investigating collision process in molecular systems. It is capable of distinguishing between solids, gases, and liquids |
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