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#77 Brownian Carnot engine-I 微觀卡諾引擎-I

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#40WBE_Scaling_Law_中一

on May 12, 2016

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#77 Brownian Carnot engine-I 微觀卡諾引擎-I

Hello everyone. Happy Chinese New Year! Today I am going to greet our readers with my humble gift, the Brownian Carnot engine model. If you are not familiar with the Carnot engine, please refer to our older post: https://goo.gl/QySuJE. /*-----------------Divider-----------------*/ The Carnot cycle proposed in 1824 sets an upper limit to the efficiency of thermal engines. However, does that still hold in the microscopic world, in which most biological motors operate? In order to experimentally study how a microscopic Carnot engine would work, the authors of our suggested reading developed a way that could effectively heat, cool, compress or expand a microsphere confined to the optical tweezers, as shown below. (figure adapted from the supplementary figure 1 of ref[1].) The optical tweezers could act as an effective spring with force constant κ, which confines the range of motion of the microsphere. Thus, by adjusting the force constant κ, we could effectively compress or expan

#75 The application of Fokker-Planck equation-II 福克-普朗克方程的應用-II

Today we are going to talk about the first passage time problem. It is important in physical, biomedical science and economy. In first passage time problem, we are interested in the quantity T(x; x 0 ) , the time it takes a particle to reach a point x from x 0. However, due to the stochastic nature of the system, T( x; x 0 ) is associated with a probability f(x,t) , which denotes the probability that x is reached at time t . We need to solve the Fokker-Planck equation for P(x,t;x0,t0). We consider only 1D system for simplicity. The boundary condition of the system looks like this: A poor drunk (so he can't recognize the cliff) random walker is walking on a 1D line with a dangerous cliff at x = x_{end} and reflecting boundary condition at x=0. We denote the probability that the dangerous cliff is not reached at time t as S(x0, t), the survival probability. From basic probability theory, the following 3 equation is obtainable: The function f is difficult to obtai

95 Fundamentals -- eigenvalue problem, Cayley-Hamilton theorem and the invariants of tensors

After previous episodes, I believe our readers have more understanding about what a tensor is. The concept of invariant is built in tensor itself -- a tensor is invariant to changes in coordination systems. However, the decomposition of a tensor $\textbf{T}$ is not invariant. $T_{ij}\neq T_{ij}'$ unless it is an isotropic tensor. Is it possible that there are something associated with a tensor that are also invariant to changes in coordination systems? That's what we are going to find out today! Eigenvalues and Eigenvectors From previous episode , we know that a second-order Cartesian tensor can a linear mapping of vectors. Usually, these vectors will change their directions after the mapping. However, there are some vectors that are kind of "intrinsic" to the tensor. These "intrinsic" vectors will not change their directions after the mapping of the tensor. We call these vectors "eigenvectors," where "eigen" means "own."

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#77 Brownian Carnot engine-I 微觀卡諾引擎-I

#40WBE_Scaling_Law_中一

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