13: Dressing QBL to a 2D square domain and constructing the effective area. Approximating the non-uniform quantum density by an effectively uniform one via the introduction of QBL.

Contexts in source publication

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... for a subatomic particle electron, one of the lightest massive particles. For atoms and molecules, the order of magnitude difference is even larger. So the essential thing separating the nano world from macro one is the comparison of the thermal de Broglie wavelength of particles with the sizes of the domain where those particles are confined. In Fig. 2.1, such a comparison is given. When L is much larger than λ th , we can assume the particles to behave as point-like. When L is comparable with λ th , wave-like behavior of particles become apparent. Hence, electrons confined in domains with nanoscale dimensions exhibit its wave nature even at room ...

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... single-layer of carbon atoms) in which electrons are free to move in two directions but restricted or confined in one direction. Accordingly, carbon nanotubes and quantum wires are considered as 1D structures that are confined in two directions. When the particles are confined in all directions, the structure is called a quantum dot, hence 0D. In Fig. 2.2, different structures of carbon-based materials confined in various directions can be seen as examples. The strength of confinement in a particular direction is determined by the confinement parameter of that ...

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... α enters as a proportionality constant of quantum states to statistical expressions of thermodynamic quantities and determines the essential discreteness in energy spectrum. Although the difference between each quantum state is Figure 2.2: Allotropes of carbon at various dimensions. ...

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... systems are the charge carriers (electrons and holes) and phonons in nanoscale metals and semiconductors in addition to ultracold atoms in optical traps. Theoretical study of these confined systems is usually done using the particle in a box model. Consider a non-relativistic single quantum particle confined in a 1D domain having length L, Fig. 2.3. The potential inside the well is zero and infinite at the outside, meaning the walls are impenetrable. Even a quantum particle cannot tunnel (leak) through the walls of an infinite well. This is a quite accurate model for the behaviors of conduction band electrons in metals for example. We will stick with the impenetrable boundaries ...

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... important thing to notice here is that energy levels are not continuous but discrete. This is one of the properties of matter which was unexplainable Figure 2.3: Stationary states correspond to standing waves in systems exhibiting wave-like behavior, if the system has a fixed geometry. ...

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... is fixed at both ends. The length of the domain is associated with the confinement of the particles. Smaller the length, higher the confinement and higher the energy of the particle by Eq. (2.5). In this regard, spatial confinement gives rise to the discreteness in momentum and energy space. Visualization of energy eigenfunctions can be seen in Fig. 2.3. In 1D systems, each mode corresponds to a different energy and a wavefunction. Modulus square of the wavefunction gives the probability of finding the particle for a given state at a certain location inside the ...

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... important consequence of the wave nature of particles is the existence of the nonzero ground state (or zero point) energy. By their very nature, wave modes and quantum state variables start from their ground state value n = 1, corresponding to their fundamental mode. In Fig. 2.4, approximations on the representation of energy spectrum of particles can be seen. Classically, energy spectrum is considered to be continuous and starts from zero. As we have seen from the particle in a box example, the true nature of the energy spectrum is discrete and it starts from a non-zero value called the ground state. At ...

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... on the keyboard, it generates different sound. The reason is you are adding another fixed node when you press anywhere on the keyboard. Although the actual length of the string doesn't change, adding another node creates two 1D domains with different lengths and they sound according to their new lengths. An example of this can also be seen in Fig. 2.5 comparing columns II and III in the 1D domain ...

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... Fig. 2.5 size characterization of domains with different dimensions is illustrated. For 3D objects, the sizes are volume, surface area, peripheral lengths and number of vertices. These are altogether named as geometric size variables. In measure theory, this definition coincides with the standard Lebesgue measure (or more generally the ...

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... an explicit example, geometric size variables of a cube are shown in Fig. 2.6. Any 3D Figure 2.4: Classical and quantum representations of the energy spectrum. Energy spectra are considered to be continuous in classical physics by the continuum approximation. For bounded systems, a better approximation called bounded continuum approximation considers the nonzero value of the ground state E 1 . Without any ...

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... an explicit example, geometric size variables of a cube are shown in Fig. 2.6. Any 3D Figure 2.4: Classical and quantum representations of the energy spectrum. ...

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... are employed in the literature. Before explaining the tools, in order to quantify our discussion and make it more physically intuitive, we'll first mention an important concept called partition function. The partition function is a powerful statistical concept that relates microscopic properties to macroscopic ones via the probability theory, Fig. 2.7. For a monatomic gas the partition function is written ...

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... show the application of the methods we'll use the partition function, since it captures the essence of the methods without loss of generality, as any other thermodynamic state function Figure 2.7: Single-particle partition function with the Boltzmann statistical weight, providing a probabilistic connection between microscopic and macroscopic properties of a system. ...

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... consists of three terms. You may have noticed that the first term actually corresponds to the continuum approximation (i.e. replacement of the sum with integral). As it is shown in Fig. 2.8, it is a reasonable approximation for systems having less significance on near ground states which corresponds to the vanishing values of the confinement parameter α. Look at the accuracy of the black curve representing the colored columns of the summations. It has been shown clearly in our other study that in order for continuum ...

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... the colored columns of the summations. It has been shown clearly in our other study that in order for continuum approximation to be used, occupation probabilities of the excited states need to be substantial. 43 If the system is so confined that its energy is close to ground state, the first term of PSF fails to represent the summation, see Fig. 2.9a. Discreteness of the sum and the smoothness of the integral can be easily noticed in Fig. 2.9a. It should be noted that the summation starts not from zero but from unity, unlike the integral. Therefore, integral representation contains the half of the zeroth value incorrectly. Bounded continuum approximation corrects this improper ...

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... in order for continuum approximation to be used, occupation probabilities of the excited states need to be substantial. 43 If the system is so confined that its energy is close to ground state, the first term of PSF fails to represent the summation, see Fig. 2.9a. Discreteness of the sum and the smoothness of the integral can be easily noticed in Fig. 2.9a. It should be noted that the summation starts not from zero but from unity, unlike the integral. Therefore, integral representation contains the half of the zeroth value incorrectly. Bounded continuum approximation corrects this improper calculation by removing the false contribution from the half of the function's zeroth value by ...

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... It should be noted that the summation starts not from zero but from unity, unlike the integral. Therefore, integral representation contains the half of the zeroth value incorrectly. Bounded continuum approximation corrects this improper calculation by removing the false contribution from the half of the function's zeroth value by the integral, Fig. 2.9b. The correction term is called the zero correction and corresponds to the second term of the PSF. Physically, zero correction excludes the false contribution of the zeroth quantum state, since there is no such state since all quantum states start from unity. All usual quantum size effect corrections to the thermodynamic properties ...

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... that integral curve passes from the middle points of the summation columns. For {i − 0.5} integral representation mistakenly calculates over the top whereas for {i + 0.5} it falls short, Fig.2.9c. In total, this surplus (green triangles) and deficiency (orange triangles) roughly compensate each other, except for strong confinements. ...

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... formulation of quantum mechanics and the understanding of geometry effects in confined systems. 225 The question of "Can one hear the shape of a drum?" that we mentioned in the introduction chapter is related with the Weyl conjecture. This question is about an inverse problem which is determining the shape of a drum from its sound, see Fig. 2.10. On the other side, the Weyl's conjecture "solves" the direct problem, which is determining the behavior of the spectrum by using its geometric properties. Of course, Weyl's conjecture cannot be used to answer Kac's question, because it is valid only at asymptotics. But this result definitely has intrigued Kac's question. In ...

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... let's see our tools in action. A visual summary of the formalism of quantum statistical thermodynamics is presented in Fig. 2.11. Consider large number of non-interacting particles confined in a domain. First, the Schrödinger equation is solved considering the boundary conditions exposed by the confinement domain, then the obtained eigenvalues are used in the appropriate partition function. In the figure, Maxwell-Boltzmann weight is used but the formalism can ...

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... stay away from the boundaries of the domain, generating a non-uniform density distribution even in thermodynamic equilibrium. Quantum boundary layer (QBL) method approximates this non-uniform density distribution with a uniform one by introducing empty layers on boundaries, thereby constituting an effective size. How this is done can be seen in Fig. 2.12. Black curve represents the exact ensemble-averaged quantum-mechanical particle number density distribution, in short we call quantum thermal density, given by the following ...

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... about the occupation probabilities of the confined particles. On the other hand, in thermodynamics, we deal with the global properties of matter. At this point, we can make an approximation by replacing the non-uniform distribution with a uniform one around the center and completely empty layer near to the boundaries, red-dashed curve in Fig. 2.12. This assumption gives us the possibility to still define global properties but also consider the wave behavior of particles. Two variables determine the thickness of the QBL, the height of the plateau (maximum density value) and the total area under the curves which has to equal to unity due to the law of conservation of ...

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... of QBL is summarized in Fig. 2.12. The partition function consists of an infinite sum of which calculation might be numerically cumbersome. By using the standard formalism of statistical mechanics, sums can be replaced by integrals, with a minor trick where the actual volume is also replaced by the effective volume which can be found by QBL formalism. Rather than ...

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... methodology is pictured more visually in Fig. 2.13. Classically, particles are uniformly distributed inside the domain, whereas in confined domains at nanoscale, they form a Figure 2.12: Quantum boundary layer (QBL) methodology in a nutshell. Ensemble-averaged quantum-mechanical density distribution of confined particles is non-uniform (black curve) and it can be approximated by a ...

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... methodology is pictured more visually in Fig. 2.13. Classically, particles are uniformly distributed inside the domain, whereas in confined domains at nanoscale, they form a Figure 2.12: Quantum boundary layer (QBL) methodology in a nutshell. ...

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... the contour plot of the 2D domain's density. The approximation of the actual density with a uniform middle region and empty near-boundary region can be seen on the rightmost subfigure of Fig. 2.13. Although QBL is constructed by considering the 1D domain, it is straightforwardly applicable to the higher dimensional domains like the one shown in Fig. ...

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... Fig. 2.14, calculation of the effective area in a 2D domain is shown. The 2D square domain has length L and QBL dictates a reduction in length by δ from every outside boundary. This makes the effective square domain's length L − 2δ. If one calculates the effective area, it can be readily found that it is A eff = L 2 − 4Lδ + 4δ 2 . Note that ...

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... its center. Classically, infinitesimally small points shouldn't change any property of the system, because they don't even have any size. However, due to the quantum nature of particles, they can feel the existence of even an infinitesimal boundary where they form a quantum boundary layer. This can be seen from the density distributions given in Fig. 2.15. For comparison, we'll execute the analysis by comparing our shape (II) with a plain square (I). Temperature is taken to be 300K for all the calculations in this subsection. Errors due to numerical calculations are ensured to be negligible by the procedures explained in Appendix A.1. We calculate the partition functions for both ...

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... second domain that we consider is the one shown in Fig. 2.16 where ITU (stands for the abbreviation of Istanbul Technical University) letters are formed by infinitesimally thin line boundaries inside a rectangle (constituting internal boundaries as opposed to the external ones which are the rectangle's boundaries). We again numerically solve the Schrödinger equation for this domain and find ...

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... of particles in this domain, QBL of external boundaries are calculated by periphery times δ, whereas QBLs of internal boundaries are twice of the periphery times δ. This is because internal boundaries lead to the evacuation of particles on both sides of the boundary, as QBL is formed in both sides of internal boundaries, unlike the external ones, Fig. 2.16. Therefore, QBL's of internal boundaries have 2δ thickness (except when they are closer to boundaries than 2δ). Periphery calculation in this domain then becomes: P = 2 × (27.5 + 17.0) + 2 × (4 × 7.0 + 5.0 + 4.5) nm (vertical lines have 7.0nm, horizontal lines have 5.0 and 4.5 nm ...

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... are almost done with the effective area calculation. But the trickiest part comes at the last. How do we calculate number of vertices in this domain with both external and internal boundaries? Should we just sum them up? The answer is no! It is clear from the density distribution in Fig. 2.16 that QBL does not form evenly among the different vertices. For example around outer corners, QBL evacuates lots of space, whereas around inner corners (e.g. the bottom of the letter U) instead of an evacuation QBL is even invited to enter more closer into the cusps. This is the case also for the open cusps (e.g. the ends of the ...

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... last domain we'd like to consider is even more complex, which is a tractricoid, see Fig. 2.17. Numerical calculations reveal that the partition function for the particles confined in this domain is ζ trac = 8.80968. It is easy to calculate the surface area and periphery for this domain. For the number of vertices, it is clear that there are four of 90 • angles (corners of the rectangle). Also, the bottom tip of the ...

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... shape might be seen as a somewhat vague concept at first thought, there are rigorous mathematical definitions of it. Here we consider the shape as an object's geometric information that is invariant under Euclidean similarity transformations such as translation, rotation, reflection and uniform scaling. 231,232 Left and right subfigures in Fig. 3.2 show the same and distinct shapes respectively. On the left table, a triangle undergoes Euclidean similarity transformations which preserves its shape. Note that they do not necessarily preserve the size, e.g. in uniform scaling. On the right table, all shapes are different than each other, although some or all of their geometric size ...

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... Fig. 5.2, T -S and τ -θ diagrams of the cycle are shown. Unlike in classical thermodynamic cycles, the work exchange during high temperature shape transformation (from steps 2 to 3) is less than that of low temperature one (from steps 4 to 1), Fig. 5.2b. Additionally, the directions of work and heat exchanges under isothermal shape transformation processes are the same, unlike the ones in isothermal volume variation, Fig. 5.2a and 5.2b. In other words, the directions of work are exactly opposite of the Stirling cycle. In the classical Stirling cycle, the isothermal expansion is ...

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... on the physical quantity that is interested in, one can probe system's free energy response to the global or local perturbations on the boundaries, which are summarized in Fig. A.2. To calculate the torque (a global effect) exerted on inner square structure, we create small angular perturbations. We look to the difference of the unperturbed and perturbed free energies with respect to the angular perturbation, which gives the torque. Similarly, a linear perturbation on all boundaries of inner square is done to ...

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... for a subatomic particle electron, one of the lightest massive particles. For atoms and molecules, the order of magnitude difference is even larger. So the essential thing separating the nano world from macro one is the comparison of the thermal de Broglie wavelength of particles with the sizes of the domain where those particles are confined. In Fig. 2.1, such a comparison is given. When L is much larger than λ th , we can assume the particles to behave as point-like. When L is comparable with λ th , wave-like behavior of particles become apparent. Hence, electrons confined in domains with nanoscale dimensions exhibit its wave nature even at room ...

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... single-layer of carbon atoms) in which electrons are free to move in two directions but restricted or confined in one direction. Accordingly, carbon nanotubes and quantum wires are considered as 1D structures that are confined in two directions. When the particles are confined in all directions, the structure is called a quantum dot, hence 0D. In Fig. 2.2, different structures of carbon-based materials confined in various directions can be seen as examples. The strength of confinement in a particular direction is determined by the confinement parameter of that ...

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... α enters as a proportionality constant of quantum states to statistical expressions of thermodynamic quantities and determines the essential discreteness in energy spectrum. Although the difference between each quantum state is Figure 2.2: Allotropes of carbon at various dimensions. ...

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... systems are the charge carriers (electrons and holes) and phonons in nanoscale metals and semiconductors in addition to ultracold atoms in optical traps. Theoretical study of these confined systems is usually done using the particle in a box model. Consider a non-relativistic single quantum particle confined in a 1D domain having length L, Fig. 2.3. The potential inside the well is zero and infinite at the outside, meaning the walls are impenetrable. Even a quantum particle cannot tunnel (leak) through the walls of an infinite well. This is a quite accurate model for the behaviors of conduction band electrons in metals for example. We will stick with the impenetrable boundaries ...

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... important thing to notice here is that energy levels are not continuous but discrete. This is one of the properties of matter which was unexplainable Figure 2.3: Stationary states correspond to standing waves in systems exhibiting wave-like behavior, if the system has a fixed geometry. ...

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... is fixed at both ends. The length of the domain is associated with the confinement of the particles. Smaller the length, higher the confinement and higher the energy of the particle by Eq. (2.5). In this regard, spatial confinement gives rise to the discreteness in momentum and energy space. Visualization of energy eigenfunctions can be seen in Fig. 2.3. In 1D systems, each mode corresponds to a different energy and a wavefunction. Modulus square of the wavefunction gives the probability of finding the particle for a given state at a certain location inside the ...

Context 42

... important consequence of the wave nature of particles is the existence of the nonzero ground state (or zero point) energy. By their very nature, wave modes and quantum state variables start from their ground state value n = 1, corresponding to their fundamental mode. In Fig. 2.4, approximations on the representation of energy spectrum of particles can be seen. Classically, energy spectrum is considered to be continuous and starts from zero. As we have seen from the particle in a box example, the true nature of the energy spectrum is discrete and it starts from a non-zero value called the ground state. At ...

Context 43

... on the keyboard, it generates different sound. The reason is you are adding another fixed node when you press anywhere on the keyboard. Although the actual length of the string doesn't change, adding another node creates two 1D domains with different lengths and they sound according to their new lengths. An example of this can also be seen in Fig. 2.5 comparing columns II and III in the 1D domain ...

Context 44

... Fig. 2.5 size characterization of domains with different dimensions is illustrated. For 3D objects, the sizes are volume, surface area, peripheral lengths and number of vertices. These are altogether named as geometric size variables. In measure theory, this definition coincides with the standard Lebesgue measure (or more generally the ...

Context 45

... an explicit example, geometric size variables of a cube are shown in Fig. 2.6. Any 3D Figure 2.4: Classical and quantum representations of the energy spectrum. Energy spectra are considered to be continuous in classical physics by the continuum approximation. For bounded systems, a better approximation called bounded continuum approximation considers the nonzero value of the ground state E 1 . Without any ...

Context 46

... an explicit example, geometric size variables of a cube are shown in Fig. 2.6. Any 3D Figure 2.4: Classical and quantum representations of the energy spectrum. ...

Context 47

... are employed in the literature. Before explaining the tools, in order to quantify our discussion and make it more physically intuitive, we'll first mention an important concept called partition function. The partition function is a powerful statistical concept that relates microscopic properties to macroscopic ones via the probability theory, Fig. 2.7. For a monatomic gas the partition function is written ...

Context 48

... show the application of the methods we'll use the partition function, since it captures the essence of the methods without loss of generality, as any other thermodynamic state function Figure 2.7: Single-particle partition function with the Boltzmann statistical weight, providing a probabilistic connection between microscopic and macroscopic properties of a system. ...

Context 49

... consists of three terms. You may have noticed that the first term actually corresponds to the continuum approximation (i.e. replacement of the sum with integral). As it is shown in Fig. 2.8, it is a reasonable approximation for systems having less significance on near ground states which corresponds to the vanishing values of the confinement parameter α. Look at the accuracy of the black curve representing the colored columns of the summations. It has been shown clearly in our other study that in order for continuum ...

Context 50

... the colored columns of the summations. It has been shown clearly in our other study that in order for continuum approximation to be used, occupation probabilities of the excited states need to be substantial. 43 If the system is so confined that its energy is close to ground state, the first term of PSF fails to represent the summation, see Fig. 2.9a. Discreteness of the sum and the smoothness of the integral can be easily noticed in Fig. 2.9a. It should be noted that the summation starts not from zero but from unity, unlike the integral. Therefore, integral representation contains the half of the zeroth value incorrectly. Bounded continuum approximation corrects this improper ...

Context 51

... in order for continuum approximation to be used, occupation probabilities of the excited states need to be substantial. 43 If the system is so confined that its energy is close to ground state, the first term of PSF fails to represent the summation, see Fig. 2.9a. Discreteness of the sum and the smoothness of the integral can be easily noticed in Fig. 2.9a. It should be noted that the summation starts not from zero but from unity, unlike the integral. Therefore, integral representation contains the half of the zeroth value incorrectly. Bounded continuum approximation corrects this improper calculation by removing the false contribution from the half of the function's zeroth value by ...

Context 52

... It should be noted that the summation starts not from zero but from unity, unlike the integral. Therefore, integral representation contains the half of the zeroth value incorrectly. Bounded continuum approximation corrects this improper calculation by removing the false contribution from the half of the function's zeroth value by the integral, Fig. 2.9b. The correction term is called the zero correction and corresponds to the second term of the PSF. Physically, zero correction excludes the false contribution of the zeroth quantum state, since there is no such state since all quantum states start from unity. All usual quantum size effect corrections to the thermodynamic properties ...

Context 53

... that integral curve passes from the middle points of the summation columns. For {i − 0.5} integral representation mistakenly calculates over the top whereas for {i + 0.5} it falls short, Fig.2.9c. In total, this surplus (green triangles) and deficiency (orange triangles) roughly compensate each other, except for strong confinements. ...

Context 54

... formulation of quantum mechanics and the understanding of geometry effects in confined systems. 225 The question of "Can one hear the shape of a drum?" that we mentioned in the introduction chapter is related with the Weyl conjecture. This question is about an inverse problem which is determining the shape of a drum from its sound, see Fig. 2.10. On the other side, the Weyl's conjecture "solves" the direct problem, which is determining the behavior of the spectrum by using its geometric properties. Of course, Weyl's conjecture cannot be used to answer Kac's question, because it is valid only at asymptotics. But this result definitely has intrigued Kac's question. In ...

Context 55

... let's see our tools in action. A visual summary of the formalism of quantum statistical thermodynamics is presented in Fig. 2.11. Consider large number of non-interacting particles confined in a domain. First, the Schrödinger equation is solved considering the boundary conditions exposed by the confinement domain, then the obtained eigenvalues are used in the appropriate partition function. In the figure, Maxwell-Boltzmann weight is used but the formalism can ...

Context 56

... stay away from the boundaries of the domain, generating a non-uniform density distribution even in thermodynamic equilibrium. Quantum boundary layer (QBL) method approximates this non-uniform density distribution with a uniform one by introducing empty layers on boundaries, thereby constituting an effective size. How this is done can be seen in Fig. 2.12. Black curve represents the exact ensemble-averaged quantum-mechanical particle number density distribution, in short we call quantum thermal density, given by the following ...

Context 57

... about the occupation probabilities of the confined particles. On the other hand, in thermodynamics, we deal with the global properties of matter. At this point, we can make an approximation by replacing the non-uniform distribution with a uniform one around the center and completely empty layer near to the boundaries, red-dashed curve in Fig. 2.12. This assumption gives us the possibility to still define global properties but also consider the wave behavior of particles. Two variables determine the thickness of the QBL, the height of the plateau (maximum density value) and the total area under the curves which has to equal to unity due to the law of conservation of ...

Context 58

... of QBL is summarized in Fig. 2.12. The partition function consists of an infinite sum of which calculation might be numerically cumbersome. By using the standard formalism of statistical mechanics, sums can be replaced by integrals, with a minor trick where the actual volume is also replaced by the effective volume which can be found by QBL formalism. Rather than ...

Context 59

... methodology is pictured more visually in Fig. 2.13. Classically, particles are uniformly distributed inside the domain, whereas in confined domains at nanoscale, they form a Figure 2.12: Quantum boundary layer (QBL) methodology in a nutshell. Ensemble-averaged quantum-mechanical density distribution of confined particles is non-uniform (black curve) and it can be approximated by a ...

Context 60

... methodology is pictured more visually in Fig. 2.13. Classically, particles are uniformly distributed inside the domain, whereas in confined domains at nanoscale, they form a Figure 2.12: Quantum boundary layer (QBL) methodology in a nutshell. ...

Context 61

... the contour plot of the 2D domain's density. The approximation of the actual density with a uniform middle region and empty near-boundary region can be seen on the rightmost subfigure of Fig. 2.13. Although QBL is constructed by considering the 1D domain, it is straightforwardly applicable to the higher dimensional domains like the one shown in Fig. ...

Context 62

... Fig. 2.14, calculation of the effective area in a 2D domain is shown. The 2D square domain has length L and QBL dictates a reduction in length by δ from every outside boundary. This makes the effective square domain's length L − 2δ. If one calculates the effective area, it can be readily found that it is A eff = L 2 − 4Lδ + 4δ 2 . Note that ...

Context 63

... its center. Classically, infinitesimally small points shouldn't change any property of the system, because they don't even have any size. However, due to the quantum nature of particles, they can feel the existence of even an infinitesimal boundary where they form a quantum boundary layer. This can be seen from the density distributions given in Fig. 2.15. For comparison, we'll execute the analysis by comparing our shape (II) with a plain square (I). Temperature is taken to be 300K for all the calculations in this subsection. Errors due to numerical calculations are ensured to be negligible by the procedures explained in Appendix A.1. We calculate the partition functions for both ...

Context 64

... second domain that we consider is the one shown in Fig. 2.16 where ITU (stands for the abbreviation of Istanbul Technical University) letters are formed by infinitesimally thin line boundaries inside a rectangle (constituting internal boundaries as opposed to the external ones which are the rectangle's boundaries). We again numerically solve the Schrödinger equation for this domain and find ...

Context 65

... of particles in this domain, QBL of external boundaries are calculated by periphery times δ, whereas QBLs of internal boundaries are twice of the periphery times δ. This is because internal boundaries lead to the evacuation of particles on both sides of the boundary, as QBL is formed in both sides of internal boundaries, unlike the external ones, Fig. 2.16. Therefore, QBL's of internal boundaries have 2δ thickness (except when they are closer to boundaries than 2δ). Periphery calculation in this domain then becomes: P = 2 × (27.5 + 17.0) + 2 × (4 × 7.0 + 5.0 + 4.5) nm (vertical lines have 7.0nm, horizontal lines have 5.0 and 4.5 nm ...

Context 66

... are almost done with the effective area calculation. But the trickiest part comes at the last. How do we calculate number of vertices in this domain with both external and internal boundaries? Should we just sum them up? The answer is no! It is clear from the density distribution in Fig. 2.16 that QBL does not form evenly among the different vertices. For example around outer corners, QBL evacuates lots of space, whereas around inner corners (e.g. the bottom of the letter U) instead of an evacuation QBL is even invited to enter more closer into the cusps. This is the case also for the open cusps (e.g. the ends of the ...

Context 67

... last domain we'd like to consider is even more complex, which is a tractricoid, see Fig. 2.17. Numerical calculations reveal that the partition function for the particles confined in this domain is ζ trac = 8.80968. It is easy to calculate the surface area and periphery for this domain. For the number of vertices, it is clear that there are four of 90 • angles (corners of the rectangle). Also, the bottom tip of the ...

Context 68

... shape might be seen as a somewhat vague concept at first thought, there are rigorous mathematical definitions of it. Here we consider the shape as an object's geometric information that is invariant under Euclidean similarity transformations such as translation, rotation, reflection and uniform scaling. 231,232 Left and right subfigures in Fig. 3.2 show the same and distinct shapes respectively. On the left table, a triangle undergoes Euclidean similarity transformations which preserves its shape. Note that they do not necessarily preserve the size, e.g. in uniform scaling. On the right table, all shapes are different than each other, although some or all of their geometric size ...

Context 69

... Fig. 5.2, T -S and τ -θ diagrams of the cycle are shown. Unlike in classical thermodynamic cycles, the work exchange during high temperature shape transformation (from steps 2 to 3) is less than that of low temperature one (from steps 4 to 1), Fig. 5.2b. Additionally, the directions of work and heat exchanges under isothermal shape transformation processes are the same, unlike the ones in isothermal volume variation, Fig. 5.2a and 5.2b. In other words, the directions of work are exactly opposite of the Stirling cycle. In the classical Stirling cycle, the isothermal expansion is ...

Context 70

... on the physical quantity that is interested in, one can probe system's free energy response to the global or local perturbations on the boundaries, which are summarized in Fig. A.2. To calculate the torque (a global effect) exerted on inner square structure, we create small angular perturbations. We look to the difference of the unperturbed and perturbed free energies with respect to the angular perturbation, which gives the torque. Similarly, a linear perturbation on all boundaries of inner square is done to ...