Differential equations are used extensively in science and engineering to model dynamical systems. The most common type is the first-order differential equation, which can model exponential growth or decay. Examples include bacterial population growth, radioactive decay, electric circuit charging/discharging, and Newton's law of cooling. Higher-order differential equations, like the second-order equation from Newton's second law, are also used to model oscillating systems like springs and pendulums. Initial conditions are needed to fully solve equations with unknown constants.
1. Einstein's theory treats atoms in a solid as independent harmonic oscillators vibrating at the same frequency. This predicts the Dulong-Petit law at high temperatures but violates experiments at low temperatures.
2. Debye improved on this by considering normal modes of vibration for the whole solid. This predicts heat capacity decreasing as T^3 at low temperatures in agreement with experiments.
3. In metals, free electrons contribute to heat capacity following Fermi-Dirac statistics. Their contribution is small except at very low temperatures where it dominates over the lattice contribution.
1) Radioactivity is the spontaneous emission of radiation by unstable atomic nuclei. It occurs as the nucleus shifts to a more stable configuration by emitting energy.
2) The principal factor determining nuclear stability is the neutron-to-proton ratio. No nucleus larger than lead-208 is stable as the strong force cannot overcome electrostatic repulsion at larger sizes.
3) The rate of radioactive decay is proportional to the number of nuclei present and follows an exponential decay model expressed as N(t)=N0e-λt, where λ is the decay constant and N0 is the initial number of nuclei.
The document discusses the kinetic model of an ideal gas from a microscopic perspective. It describes the assumptions of the model, which include molecules behaving as point particles that do not interact and obey Newton's laws. It then outlines the steps to relate the microscopic description to the macroscopic ideal gas law: (1) calculating molecule collisions with the wall, (2) resulting momentum change, (3) force exerted by the wall, (4) force exerted on the wall using Newton's 3rd law, (5) defining pressure, and (6) relating this to the ideal gas law. It further defines temperature microscopically in terms of molecular kinetic energy.
I am Samantha K. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, McGill University, Canada
I have been helping students with their homework for the past 8 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
1. The document discusses differential equations and their applications. It defines differential equations and describes their use in fields like physics, engineering, biology and economics to model complex systems.
2. Examples of first order differential equations are given to model exponential growth, exponential decay, and an RL circuit. Higher order differential equations are used to model falling objects and Newton's law of cooling.
3. The key applications covered are population growth, radioactive decay, free falling objects, heat transfer, and electric circuits. Solving the differential equations gives mathematical models relating variables like position, temperature, and current over time.
Differential equations model real-world phenomena involving continuously changing quantities and their rates of change. Some examples include:
1) Population growth modeled by an exponential growth differential equation where the rate of change of population is proportional to the current population.
2) The motion of a falling object modeled by a differential equation where acceleration due to gravity relates the rate of change of velocity to the rate of change of height over time.
3) Newton's law of cooling modeled by a differential equation where the rate of change of temperature is proportional to the difference between the temperature of an object and its environment.
4) The electric current in an RL circuit modeled by a differential equation relating the rate of change of current to
This document contains a professor's problem set assignment on advanced differential equations for EGR 509. It includes 9 problems covering various types of differential equations like population growth, evaporation rates, tumor growth, Bernoulli's equation, Riccati equations, Newton's law of cooling, light absorption, drug concentration modeling, and a partial differential equation. The problems have analytical solutions provided in the form of equations.
lecture pf control system_thermal system_206.pdfAtmacaDevrim
The document discusses thermal systems and concepts such as:
- Thermal systems involve the storage and transfer of energy as heat. Heat flows from higher to lower temperatures.
- The law of conservation of energy applies to thermal systems, where the change in internal energy equals heat supplied minus work done.
- Thermal resistance and capacitance relate temperature differences and heat flow in thermal systems, analogous to voltage and capacitance in electrical systems.
- Heat transfer occurs through conduction, convection and radiation, and can be modeled using concepts like Newton's law of cooling.
1. Einstein's theory treats atoms in a solid as independent harmonic oscillators vibrating at the same frequency. This predicts the Dulong-Petit law at high temperatures but violates experiments at low temperatures.
2. Debye improved on this by considering normal modes of vibration for the whole solid. This predicts heat capacity decreasing as T^3 at low temperatures in agreement with experiments.
3. In metals, free electrons contribute to heat capacity following Fermi-Dirac statistics. Their contribution is small except at very low temperatures where it dominates over the lattice contribution.
1) Radioactivity is the spontaneous emission of radiation by unstable atomic nuclei. It occurs as the nucleus shifts to a more stable configuration by emitting energy.
2) The principal factor determining nuclear stability is the neutron-to-proton ratio. No nucleus larger than lead-208 is stable as the strong force cannot overcome electrostatic repulsion at larger sizes.
3) The rate of radioactive decay is proportional to the number of nuclei present and follows an exponential decay model expressed as N(t)=N0e-λt, where λ is the decay constant and N0 is the initial number of nuclei.
The document discusses the kinetic model of an ideal gas from a microscopic perspective. It describes the assumptions of the model, which include molecules behaving as point particles that do not interact and obey Newton's laws. It then outlines the steps to relate the microscopic description to the macroscopic ideal gas law: (1) calculating molecule collisions with the wall, (2) resulting momentum change, (3) force exerted by the wall, (4) force exerted on the wall using Newton's 3rd law, (5) defining pressure, and (6) relating this to the ideal gas law. It further defines temperature microscopically in terms of molecular kinetic energy.
I am Samantha K. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, McGill University, Canada
I have been helping students with their homework for the past 8 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
1. The document discusses differential equations and their applications. It defines differential equations and describes their use in fields like physics, engineering, biology and economics to model complex systems.
2. Examples of first order differential equations are given to model exponential growth, exponential decay, and an RL circuit. Higher order differential equations are used to model falling objects and Newton's law of cooling.
3. The key applications covered are population growth, radioactive decay, free falling objects, heat transfer, and electric circuits. Solving the differential equations gives mathematical models relating variables like position, temperature, and current over time.
Differential equations model real-world phenomena involving continuously changing quantities and their rates of change. Some examples include:
1) Population growth modeled by an exponential growth differential equation where the rate of change of population is proportional to the current population.
2) The motion of a falling object modeled by a differential equation where acceleration due to gravity relates the rate of change of velocity to the rate of change of height over time.
3) Newton's law of cooling modeled by a differential equation where the rate of change of temperature is proportional to the difference between the temperature of an object and its environment.
4) The electric current in an RL circuit modeled by a differential equation relating the rate of change of current to
This document contains a professor's problem set assignment on advanced differential equations for EGR 509. It includes 9 problems covering various types of differential equations like population growth, evaporation rates, tumor growth, Bernoulli's equation, Riccati equations, Newton's law of cooling, light absorption, drug concentration modeling, and a partial differential equation. The problems have analytical solutions provided in the form of equations.
lecture pf control system_thermal system_206.pdfAtmacaDevrim
The document discusses thermal systems and concepts such as:
- Thermal systems involve the storage and transfer of energy as heat. Heat flows from higher to lower temperatures.
- The law of conservation of energy applies to thermal systems, where the change in internal energy equals heat supplied minus work done.
- Thermal resistance and capacitance relate temperature differences and heat flow in thermal systems, analogous to voltage and capacitance in electrical systems.
- Heat transfer occurs through conduction, convection and radiation, and can be modeled using concepts like Newton's law of cooling.
Chapter 6 Thermally Activated Process and Diffusion in Solids.Pem(ເປ່ມ) PHAKVISETH
This document discusses thermally activated processes and diffusion in solids. It covers topics like rate processes, the probability of atoms acquiring activation energy, vacancy and interstitial diffusion mechanisms, and industrial applications of diffusion like surface hardening and integrated circuits. The key points are:
- Reactions in solids require atoms to gain enough energy to overcome activation energy barriers. Higher temperatures provide more energy.
- Diffusion occurs through vacancy or interstitial mechanisms as atoms move into vacant spaces in the crystal lattice. It is described by equations like Fick's laws.
- Industrial uses of diffusion include carburizing steel surfaces to introduce carbon and dope silicon wafers with impurities to create integrated circuits. Diffusion
Nuclear Decay - A Mathematical PerspectiveErik Faust
Radioactivity as a phenomenon is often misunderstood: if one says ‘Radioactive’, most people will think about disastrous electrical plants, dangerous bombs and other forms of life-threatening details. In my native Germany, members of the Green party have been campaigning for a decade to put an end to nuclear energy. Only few think of the useful aspects of this unique actuality, although radiotherapy is most promising of tools in the fight against cancer, and radioactive dating allows us to identify the age of any historical item. But even fewer people see radioactivity as the natural process that it actually is: A spontaneous mechanism, in which one nucleus decays into another. As an aspiring Physicist and Engineer, Radioactivity is one my favourite topics in the realm of science. I am fascinated at how we are able to predict exactly how many Nuclei will decay in a certain amount of time, but not say for certain which Nuclei exactly will do so.
This document discusses chemical kinetics and reaction rates. It explains that kinetics studies how fast chemical reactions occur. The rate of a reaction depends on factors like the concentrations of reactants, temperature, and presence of catalysts. Reaction rates can be determined by measuring changes in concentration over time. The order of a reaction indicates how the rate depends on reactant concentrations. First-order and second-order reactions follow distinct rate laws that allow calculation of rate constants from experimental data. Reaction mechanisms involve elementary steps that may be fast or slow, with the overall rate determined by the slowest step.
This document discusses chemical kinetics and reaction rates. It explains that kinetics studies how fast chemical reactions occur. The rate of a reaction depends on factors like the concentrations of reactants, temperature, and presence of catalysts. Reaction rates can be determined by measuring changes in concentration over time. The order of a reaction indicates how the rate depends on reactant concentrations. First-order and second-order reactions follow distinct rate laws that allow calculation of rate constants from experimental data. Reaction mechanisms involve elementary steps that describe the pathway by which reactants are converted to products.
APPLICATION OF HIGHER ORDER DIFFERENTIAL EQUATIONSAYESHA JAVED
1) The document discusses modeling and applications of second order differential equations. It provides examples of second order differential equations that model vibrating springs and electric current circuits.
2) Solving second order differential equations involves finding the complementary function and particular integral. The type of roots in the auxiliary equation determines the form of the complementary function.
3) An example solves a second order differential equation modeling a vibrating spring to find the position of a mass attached to the spring at any time.
Fi ck law
Diffusion: random walk of an ensemble of particles from region of high “concentration” to region of small “concentration”.
Flow is proportional to the negative gradient of the “concentration”.
The document discusses the concept of entropy. It begins by defining entropy as a measure of disorder in a system and relating it to the number of microscopic states available to a system. It then discusses different types of disorder, provides an example, and gives mathematical expressions to define and calculate entropy. The document notes the importance of entropy in geochemical thermodynamics and applications like thermobarometric models. It concludes by restating that entropy increases with heat transfer at low temperatures and decreases with heat removal, and is a key concept in understanding the direction of natural processes.
The document discusses the concept of entropy. It begins by defining entropy as a measure of disorder in a system and relating it to the number of microscopic states available to a system. It then discusses different types of disorder, provides an example, and gives mathematical expressions to define and calculate entropy. The document notes the importance of entropy in geochemical thermodynamics and applications like thermobarometric models. It concludes by restating that entropy increases with heat transfer at low temperatures and decreases with heat removal, and is a key concept in understanding the direction of natural processes.
The document discusses a lecture on statistical thermodynamics. It introduces the concept of a partition function, which describes the possible energy states of a system and the probability of occupying those states. It provides examples of using the Boltzmann distribution and Lagrange multipliers to determine the most probable distribution of molecules among energy levels for different systems. The summary focuses on key statistical thermodynamics concepts introduced in the document.
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Chemical kinematics deals with the rates of chemical reactions and their mechanisms. The rate constant in a rate law, such as the rate law d[A]/dt = k[A], is independent of concentration but depends on factors like temperature. The order of a reaction is the sum of the powers of the concentration terms in its rate equation. Common orders of reaction include zero-order, where the rate is independent of reactant concentration, first-order, where the rate depends on one concentration term, and second-order, where the rate depends on two concentration terms. Higher order reactions are also possible but are more complex.
The document discusses Boltzmann statistics and systems in thermal contact with heat reservoirs. It begins by stating the fundamental assumption for isolated systems, which is that all microstates are equally probable. It then discusses how to apply this to systems in contact with heat reservoirs. The key results are:
1) For a system in contact with a heat reservoir at temperature T, the probability of the system being in a microstate with energy ε is proportional to the Boltzmann factor e^(-ε/kT).
2) The ratio of probabilities of two microstates with energies ε1 and ε2 is given by the ratio of their Boltzmann factors.
3) Systems visited microstates with
Einstein published five groundbreaking papers in 1905, including his paper on Brownian motion. In this paper, Einstein analyzed the random movement of microscopic particles suspended in a fluid using the kinetic theory of gases and statistical mechanics. He determined that the diffusion and irregular motion of particles is caused by bombardment by the atoms/molecules in the surrounding fluid. This provided direct evidence for the atomic theory that was still controversial at the time. Einstein's analysis reconciled the kinetic theory with the second law of thermodynamics and allowed physicists to understand Brownian motion at the atomic scale through statistical analysis.
Phase-field modeling of crystal nucleation I: Fundamentals and methodsPFHub PFHub
This document summarizes phase-field modeling of homogeneous crystal nucleation using two main methods. The first method adds fluctuations (noise) to the phase-field equations of motion to mimic natural nucleation. The noise amplitude is determined by the fluctuation-dissipation theorem. This models nucleation without assuming a sharp interface or bulk properties. The second method places supercritical crystal seeds randomly in space and time to model nucleation. Quantitative results from both methods are difficult to obtain due to limitations of classical nucleation theory. The document outlines the phase-field model and equations used, including discretization for simulation. It demonstrates convergence of results with decreasing grid spacing and time step when modeling crystal growth and nucleation with noise.
Phase-field modeling of crystal nucleation I: Fundamentals and methodsDaniel Wheeler
This document summarizes phase-field modeling of homogeneous crystal nucleation using two main methods. The first method adds fluctuations (noise) to the phase-field equations of motion to mimic natural nucleation. The noise amplitude is determined by the fluctuation-dissipation theorem. This models nucleation without assuming a sharp interface or bulk properties. The second method places supercritical crystal seeds randomly in space and time to model nucleation. Quantitative results from both methods are difficult to obtain due to limitations of classical nucleation theory. The document outlines the phase-field model and equations used to simulate homogeneous nucleation and crystal growth in 2D with and without noise to demonstrate convergence with spatial and temporal discretization.
The document discusses the effective mass approximation in quantum mechanics. It begins by defining the effective mass as inversely proportional to the curvature of energy bands. Having a effective mass allows electrons in crystals to be treated similarly to classical particles, with the crystal forces and quantum properties accounted for in the mass. The effective mass can be a tensor and depends on the crystal direction. It then discusses measuring the effective mass using cyclotron resonance and how it varies by crystallographic direction. In general, the effective mass incorporates the quantum mechanical behavior of electrons in crystals to allow a classical particle treatment.
Thermal diffusivity describes how quickly heat diffuses through a material. It is calculated as the thermal conductivity divided by the density and specific heat. Fick's laws of diffusion quantitatively describe steady-state and non-steady-state diffusion. For a heat pulse experiment passing through a brass tube, the temperature was measured at two points over time. Fourier analysis was used to determine the amplitude and phase of the temperature waves. The ratio of amplitudes and difference in phases was used to calculate the thermal diffusivity, found to be 0.231 cm^2/s, close to the actual value for brass of 0.3 cm^2/s.
The passage discusses the importance of summarization in an age of information overload. It notes that with the massive amount of online information available, being able to quickly understand the key points of documents is crucial. The ability to produce concise yet informative summaries can help people navigate large amounts of content and identify what is most relevant or important to their needs.
Pin By Rhonda Genusa On Writing Process Teaching Writing, WritingJeff Nelson
The document discusses the contrasting philosophies of W.E.B. Du Bois and Booker T. Washington regarding the best approach for African Americans to overcome racial discrimination after the Civil War. Du Bois advocated for increased access to education and political rights, while Washington believed African Americans should focus first on industrial education and economic empowerment through occupations like farming and domestic work. Both men aimed to uplift the black community, but had differing views on the path forward.
In The Great Gatsby, F. Scott Fitzgerald uses the color green to symbolize wealth, aspirations for the American Dream, and the pursuit of happiness. Green is prominently featured in descriptions of Jay Gatsby's lavish parties and mansion, representing his immense fortune and desire to attain status. The novel suggests that true happiness cannot be bought, as represented by the fading of green at the end of the story along with Gatsby's dreams.
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Semelhante a 3 Applications Of Differential Equations
Chapter 6 Thermally Activated Process and Diffusion in Solids.Pem(ເປ່ມ) PHAKVISETH
This document discusses thermally activated processes and diffusion in solids. It covers topics like rate processes, the probability of atoms acquiring activation energy, vacancy and interstitial diffusion mechanisms, and industrial applications of diffusion like surface hardening and integrated circuits. The key points are:
- Reactions in solids require atoms to gain enough energy to overcome activation energy barriers. Higher temperatures provide more energy.
- Diffusion occurs through vacancy or interstitial mechanisms as atoms move into vacant spaces in the crystal lattice. It is described by equations like Fick's laws.
- Industrial uses of diffusion include carburizing steel surfaces to introduce carbon and dope silicon wafers with impurities to create integrated circuits. Diffusion
Nuclear Decay - A Mathematical PerspectiveErik Faust
Radioactivity as a phenomenon is often misunderstood: if one says ‘Radioactive’, most people will think about disastrous electrical plants, dangerous bombs and other forms of life-threatening details. In my native Germany, members of the Green party have been campaigning for a decade to put an end to nuclear energy. Only few think of the useful aspects of this unique actuality, although radiotherapy is most promising of tools in the fight against cancer, and radioactive dating allows us to identify the age of any historical item. But even fewer people see radioactivity as the natural process that it actually is: A spontaneous mechanism, in which one nucleus decays into another. As an aspiring Physicist and Engineer, Radioactivity is one my favourite topics in the realm of science. I am fascinated at how we are able to predict exactly how many Nuclei will decay in a certain amount of time, but not say for certain which Nuclei exactly will do so.
This document discusses chemical kinetics and reaction rates. It explains that kinetics studies how fast chemical reactions occur. The rate of a reaction depends on factors like the concentrations of reactants, temperature, and presence of catalysts. Reaction rates can be determined by measuring changes in concentration over time. The order of a reaction indicates how the rate depends on reactant concentrations. First-order and second-order reactions follow distinct rate laws that allow calculation of rate constants from experimental data. Reaction mechanisms involve elementary steps that may be fast or slow, with the overall rate determined by the slowest step.
This document discusses chemical kinetics and reaction rates. It explains that kinetics studies how fast chemical reactions occur. The rate of a reaction depends on factors like the concentrations of reactants, temperature, and presence of catalysts. Reaction rates can be determined by measuring changes in concentration over time. The order of a reaction indicates how the rate depends on reactant concentrations. First-order and second-order reactions follow distinct rate laws that allow calculation of rate constants from experimental data. Reaction mechanisms involve elementary steps that describe the pathway by which reactants are converted to products.
APPLICATION OF HIGHER ORDER DIFFERENTIAL EQUATIONSAYESHA JAVED
1) The document discusses modeling and applications of second order differential equations. It provides examples of second order differential equations that model vibrating springs and electric current circuits.
2) Solving second order differential equations involves finding the complementary function and particular integral. The type of roots in the auxiliary equation determines the form of the complementary function.
3) An example solves a second order differential equation modeling a vibrating spring to find the position of a mass attached to the spring at any time.
Fi ck law
Diffusion: random walk of an ensemble of particles from region of high “concentration” to region of small “concentration”.
Flow is proportional to the negative gradient of the “concentration”.
The document discusses the concept of entropy. It begins by defining entropy as a measure of disorder in a system and relating it to the number of microscopic states available to a system. It then discusses different types of disorder, provides an example, and gives mathematical expressions to define and calculate entropy. The document notes the importance of entropy in geochemical thermodynamics and applications like thermobarometric models. It concludes by restating that entropy increases with heat transfer at low temperatures and decreases with heat removal, and is a key concept in understanding the direction of natural processes.
The document discusses the concept of entropy. It begins by defining entropy as a measure of disorder in a system and relating it to the number of microscopic states available to a system. It then discusses different types of disorder, provides an example, and gives mathematical expressions to define and calculate entropy. The document notes the importance of entropy in geochemical thermodynamics and applications like thermobarometric models. It concludes by restating that entropy increases with heat transfer at low temperatures and decreases with heat removal, and is a key concept in understanding the direction of natural processes.
The document discusses a lecture on statistical thermodynamics. It introduces the concept of a partition function, which describes the possible energy states of a system and the probability of occupying those states. It provides examples of using the Boltzmann distribution and Lagrange multipliers to determine the most probable distribution of molecules among energy levels for different systems. The summary focuses on key statistical thermodynamics concepts introduced in the document.
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Chemical kinematics deals with the rates of chemical reactions and their mechanisms. The rate constant in a rate law, such as the rate law d[A]/dt = k[A], is independent of concentration but depends on factors like temperature. The order of a reaction is the sum of the powers of the concentration terms in its rate equation. Common orders of reaction include zero-order, where the rate is independent of reactant concentration, first-order, where the rate depends on one concentration term, and second-order, where the rate depends on two concentration terms. Higher order reactions are also possible but are more complex.
The document discusses Boltzmann statistics and systems in thermal contact with heat reservoirs. It begins by stating the fundamental assumption for isolated systems, which is that all microstates are equally probable. It then discusses how to apply this to systems in contact with heat reservoirs. The key results are:
1) For a system in contact with a heat reservoir at temperature T, the probability of the system being in a microstate with energy ε is proportional to the Boltzmann factor e^(-ε/kT).
2) The ratio of probabilities of two microstates with energies ε1 and ε2 is given by the ratio of their Boltzmann factors.
3) Systems visited microstates with
Einstein published five groundbreaking papers in 1905, including his paper on Brownian motion. In this paper, Einstein analyzed the random movement of microscopic particles suspended in a fluid using the kinetic theory of gases and statistical mechanics. He determined that the diffusion and irregular motion of particles is caused by bombardment by the atoms/molecules in the surrounding fluid. This provided direct evidence for the atomic theory that was still controversial at the time. Einstein's analysis reconciled the kinetic theory with the second law of thermodynamics and allowed physicists to understand Brownian motion at the atomic scale through statistical analysis.
Phase-field modeling of crystal nucleation I: Fundamentals and methodsPFHub PFHub
This document summarizes phase-field modeling of homogeneous crystal nucleation using two main methods. The first method adds fluctuations (noise) to the phase-field equations of motion to mimic natural nucleation. The noise amplitude is determined by the fluctuation-dissipation theorem. This models nucleation without assuming a sharp interface or bulk properties. The second method places supercritical crystal seeds randomly in space and time to model nucleation. Quantitative results from both methods are difficult to obtain due to limitations of classical nucleation theory. The document outlines the phase-field model and equations used, including discretization for simulation. It demonstrates convergence of results with decreasing grid spacing and time step when modeling crystal growth and nucleation with noise.
Phase-field modeling of crystal nucleation I: Fundamentals and methodsDaniel Wheeler
This document summarizes phase-field modeling of homogeneous crystal nucleation using two main methods. The first method adds fluctuations (noise) to the phase-field equations of motion to mimic natural nucleation. The noise amplitude is determined by the fluctuation-dissipation theorem. This models nucleation without assuming a sharp interface or bulk properties. The second method places supercritical crystal seeds randomly in space and time to model nucleation. Quantitative results from both methods are difficult to obtain due to limitations of classical nucleation theory. The document outlines the phase-field model and equations used to simulate homogeneous nucleation and crystal growth in 2D with and without noise to demonstrate convergence with spatial and temporal discretization.
The document discusses the effective mass approximation in quantum mechanics. It begins by defining the effective mass as inversely proportional to the curvature of energy bands. Having a effective mass allows electrons in crystals to be treated similarly to classical particles, with the crystal forces and quantum properties accounted for in the mass. The effective mass can be a tensor and depends on the crystal direction. It then discusses measuring the effective mass using cyclotron resonance and how it varies by crystallographic direction. In general, the effective mass incorporates the quantum mechanical behavior of electrons in crystals to allow a classical particle treatment.
Thermal diffusivity describes how quickly heat diffuses through a material. It is calculated as the thermal conductivity divided by the density and specific heat. Fick's laws of diffusion quantitatively describe steady-state and non-steady-state diffusion. For a heat pulse experiment passing through a brass tube, the temperature was measured at two points over time. Fourier analysis was used to determine the amplitude and phase of the temperature waves. The ratio of amplitudes and difference in phases was used to calculate the thermal diffusivity, found to be 0.231 cm^2/s, close to the actual value for brass of 0.3 cm^2/s.
The passage discusses the importance of summarization in an age of information overload. It notes that with the massive amount of online information available, being able to quickly understand the key points of documents is crucial. The ability to produce concise yet informative summaries can help people navigate large amounts of content and identify what is most relevant or important to their needs.
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The document discusses the contrasting philosophies of W.E.B. Du Bois and Booker T. Washington regarding the best approach for African Americans to overcome racial discrimination after the Civil War. Du Bois advocated for increased access to education and political rights, while Washington believed African Americans should focus first on industrial education and economic empowerment through occupations like farming and domestic work. Both men aimed to uplift the black community, but had differing views on the path forward.
In The Great Gatsby, F. Scott Fitzgerald uses the color green to symbolize wealth, aspirations for the American Dream, and the pursuit of happiness. Green is prominently featured in descriptions of Jay Gatsby's lavish parties and mansion, representing his immense fortune and desire to attain status. The novel suggests that true happiness cannot be bought, as represented by the fading of green at the end of the story along with Gatsby's dreams.
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- Data centers require massive servers, storage, networking equipment and facilities to power millions of devices and store/process huge amounts of user data.
- By building large data centers, Apple can handle the growing demand for its services like iCloud, App Store, Apple Music etc. as more people use Apple devices and subscribe to its services.
- The location in North Carolina provides benefits like access to renewable energy sources, tax incentives from the state government and a suitable climate for data center operations.
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Here are a few reflections on your experience in PE class:
- You showed determination and perseverance by pushing through the physical activities despite hurting your toe over the weekend. Completing the exercises in spite of discomfort demonstrates strength of character.
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- Having a specific goal of doing your best physically helped give direction and motivation. Setting intentions, then reviewing experiences against them, provides perspective and opportunities for learning.
- While some parts were challenging, you found success in other
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
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Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
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it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
1. 3 Applications of Differential Equations
Differential equations are absolutely fundamental to modern science and engineering.
Almost all of the known laws of physics and chemistry are actually differential equa-
tions, and differential equation models are used extensively in biology to study bio-
A mathematical model is a
description of a real-world system
using mathematical language and
ideas.
chemical reactions, population dynamics, organism growth, and the spread of diseases.
The most common use of differential equations in science is to model dynamical
systems, i.e. systems that change in time according to some fixed rule. For such a
system, the independent variable is t (for time) instead of x, meaning that equations
are written like
dy
dt
= t3
y2
instead of y′
= x3
y2
.
In addition, the letter y is usually replaced by a letter that represents the variable
under consideration, e.g. M for mass, P for population, T for temperature, and so
forth.
Exponential Growth and Decay
Perhaps the most common differential equation in the sciences is the following.
THE NATURAL GROWTH EQUATION
The natural growth equation is the differential equation
dy
dt
= ky
where k is a constant. Its solutions have the form
y = y0 ekt
where y0 = y(0) is the initial value of y.
The constant k is called the rate constant or growth constant, and has units of
inverse time (number per second). The sign of k governs the behavior of the solutions:
• If k > 0, then the variable y increases exponentially over time. This is called
exponential growth.
• If k < 0, then the variable y decreases over time, approaching zero asymptotically.
This is called exponential decay.
See Figure 1 for sample graphs of y = ekt
in these two cases. In the case where k is
k > 0
t
y
y = ekt
k < 0
t
y
y = ekt
Figure 1: Exponential growth
and decay.
negative, the natural growth equation can also be written
dy
dt
= −ry
where r = |k| is positive, in which case the solutions have the form y = y0e−rt
.
The following examples illustrate several instances in science where exponential
growth or decay is relevant.
EXAMPLE 1 Consider a colony of bacteria in a resource-rich environment. Here
“resource-rich” means, for example, that there is plenty of food, as well as space for
2. APPLICATIONS OF DIFFERENTIAL EQUATIONS 2
the colony to grow. In such an environment, the population P of the colony will grow,
as individual bacteria reproduce via binary fission.
Assuming that no bacteria die, the rate at which such a population grows will be
proportional to the number of bacteria. For example, the population might increase at
a rate of 5% per minute, regardless of its size. Intuitively, this is because the rate at
which individual bacterial cells divide does not depend on the number of cells.
We can express this rule as a differential equation:
dP
dt
= kP.
Here k is a constant of proportionality, which can be interpreted as the rate at which
the bacteria reproduce. For example, if k = 3/hour, it means that each individual
bacteria cell has an average of 3 offspring per hour (not counting grandchildren).
It follows that the population of bacteria will grow exponentially with time:
P = P0 ekt
where P0 is the population at time t = 0 (see Figure 2).
t
P
P = P0ekt
P0
Figure 2: Exponential growth of
a bacteria population. EXAMPLE 2 Consider a sample of a certain radioactive isotope. The atoms of such
an isotope are unstable, with a certain proportion decaying each second. In particular,
the mass M of the sample will decrease as atoms are lost, with the rate of decrease
proportional to the number of atoms. We can write this as a differential equation
dM
dt
= −rM,
where r is a constant of proportionality. It follows that the mass of the sample will
decay exponentially with time:
M = M0 e−rt
,
t
M
M = M0e-rt
M0
Figure 3: Exponential decay of a
radioactive isotope.
where M0 is the mass of the sample at time t = 0 (see Figure 2).
One important measure of the rate of exponential decay is the half life. Given a
decaying variable
y = y0 e−rt
(r 0)
the half life is the amount of time that it takes for y to decrease to half of its original
value. The half life can be obtained by substituting y = y0/2
y0
2
= y0 e−rt
and then solving for t.
Similarly, given a growing variable
y = y0 ekt
(k 0)
we can measure the rate of exponential growth using the doubling time, i.e. the
amount of time that it takes for y to grow to twice its original value. The doubling
time can be obtained by substituting y = 2y0 and then solving for t.
The following example illustrates a more complicated situation where the natural
growth equation arises.
EXAMPLE 3 Figure 4 shows a simple kind of electric circuit known as an RC circuit.
This circuit has two components:
3. APPLICATIONS OF DIFFERENTIAL EQUATIONS 3
• A resistor is any circuit component—such as a light bulb—that resists the flow
of electric charge. Resistors obey Ohm’s law
V = IR,
where V is the voltage applied to the resistor, I is the rate at which charge flows
through the resistor, and R is a constant called the resistance.
• A capacitor is a circuit component that stores a supply of electric charge. When
capacitor
resistor
Figure 4: An RC circuit.
it is attached to a resistor, the capacitor will push this charge through the resistor,
creating electric current. Capacitors obey the equation
V =
Q
C
,
where Q is the charge stored in the capacitor, C is a constant called the capac-
itance of the capacitor, and V is the resulting voltage.
In an RC circuit, the voltage produced by a capacitor is applied directly across a
resistor. Setting the two formulas for V equal to each other gives
IR =
Q
C
.
Moreover, the rate I at which charge flows through the resistor is the same as the rate
at which charge flows out of the capacitor, so
I = −
dQ
dt
.
Putting these together gives the differential equation
−
dQ
dt
R =
Q
C
,
or equivalently
dQ
dt
= −
1
RC
Q.
It follows that the amount of charge held in the capacitor will decay exponentially over
time
Q = Q0 e−rt
where r = 1/(RC). In the case where the resistor is a light bulb, this means that
Although the light bulb will
technically never go out, in reality the
light will become too faint to see after
a short time.
the bulb will become dimmer and dimmer over time, although it will never quite go
out.
Separation of Variables
Many differential equations in science are separable, which makes it easy to find a
solution.
EXAMPLE 4 Newton’s law of cooling is a differential equation that predicts the
cooling of a warm body placed in a cold environment. According to the law, the rate
at which the temperature of the body decreases is proportional to the difference of
temperature between the body and its environment. In symbols
dT
dt
= −k(T − Te),
4. APPLICATIONS OF DIFFERENTIAL EQUATIONS 4
where T is the temperature of the object, Te is the (constant) temperature of the
environment, and k is a constant of proportionality.
We can solve this differential equation using separation of variables. We get
Z
dT
T − Te
=
Z
−k dt,
so
ln |T − Te| = −kt + C.
Solving for T gives an equation of the form
T = Te + Ce−kt
t
T
T = Te + Ce-kt
T0
Te
Figure 5: Cooling of a warm
body.
where the value of C changed. This function decreases exponentially, but approaches Te
as t → ∞ instead of zero (see Figure 5).
EXAMPLE 5 In chemistry, the rate at which a given chemical reaction occurs is
often determined by a differential equation. For example, consider the decomposition
of nitrogen dioxide:
2 NO2 −→ 2 NO + O2.
Because this reaction requires two molecules of NO2, the rate at which the reaction
occurs is proportional to the square of the concentration of NO2. That is,
We are using the usual chemistry
notation, where [NO2] denotes the
concentration of NO2. An alternative
would be to use a single letter for this
concentration, such as N.
d[NO2]
dt
= −k[NO2]2
where [NO2] is the concentration of NO2, and k is a constant.
We can solve this equation using separation of variables. We get
Z
[NO2]−2
d[NO2] =
Z
−k dt
so
−[NO2]−1
= −kt + C.
Solving for [NO2] gives
[NO2] =
1
kt + C
t
@NO2D
@NO2D =
1
kt + C
Figure 6: Decomposition of NO2. where the value of C changed. An example graph corresponding to this formula is
shown in Figure 6. Unlike exponential decay, the concentration decreases very quickly
at first, but then very slowly afterwards.
EXAMPLE 6 Consider a colony of bacteria growing in an environment with limited
resources. For example, there may be a scarcity of food, or space constraints on the
size of the colony. In this case, it is not reasonable to expect the colony to grow
exponentially—indeed, the colony will unable to grow larger than some maximum
population Pmax.
The maximum population Pmax is
called the carrying capacity of the
bacteria colony in the given
environment.
In this case, a common model for the growth of the colony is the logistic equation
dP
dt
= kP
1 −
P
Pmax
Here the factor of 1 − P/Pmax is unimportant when P is small, but when P is close to
Pmax this factor decreases the rate of growth. Indeed, in the case where P = Pmax, this
factor forces dP/dt to be zero, meaning that the colony does not grow at all.
5. APPLICATIONS OF DIFFERENTIAL EQUATIONS 5
We can solve this differential equation using separation of variables, though it is a
bit difficult. We begin by multiplying through by Pmax
Pmax
dP
dt
= kP(Pmax − P).
We can now separate to get
Z
Pmax
P(Pmax − P)
dP =
Z
k dt.
The integral on the left is difficult to evaluate. The secret is to express the fraction as
the sum of two simpler fractions:
Pmax
P(Pmax − P)
=
1
P
+
1
Pmax − P
.
This is a simple example of the
integration technique known as
partial fractions decomposition. Each of the simpler fractions can then be integrated easily. The result is
ln |P| − ln |Pmax − P| = kt + C.
We can use a logarithm rule to combine the two terms on the left:
ln
P
Pmax − P
= kt + C
so
P
Pmax − P
= Cekt
.
Solving for P gives
P =
Pmax
1 + Ce−kt
where the value of C changed.
t
P
P =
Pmax
1 + Ce-kt
Pmax
Figure 7: Logistic population
growth.
Figure 7 shows the graph of a typical solution. Note that the population grows
quickly at first, but the rate of increase slows as the population reaches the maximum.
As t → ∞, the population asymptotically approaches Pmax.
In many of the examples we have seen, a differential equation includes an unknown
constant k. This means that the general solution will involve two unknown constants
(k and C). To solve such an equation, you will need two pieces of information, such as
the values of y(0) and y′
(0), or two different values of y.
The following example illustrates this procedure.
EXAMPLE 7 An apple pie with an initial temperature of 170 ◦
C is removed from
the oven and left to cool in a room with an air temperature of 20 ◦
C. Given that the
temperature of the pie initially decreases at a rate of 3.0 ◦
C/min, how long will it take
for the pie to cool to a temperature of 30 ◦
C?
SOLUTION Assuming the pie obeys Newton’s law of cooling (see Example 4), we have
the following information:
dT
dt
= −k(T − 20), T(0) = 170, T′
(0) = −3.0,
where T is the temperature of the pie in celsius, t is the time in minutes, and k is an
unknown constant.
We can easily find the value of k by plugging the information we know about t = 0
directly into the differential equation:
−2.5 = −k(170 − 20).
6. APPLICATIONS OF DIFFERENTIAL EQUATIONS 6
Second-Order Equations
Although first-order equations are the most common type in chemistry and biology, in
physics most systems are modeled using second-order equations. This is because of Newton’s
second law:
F = ma.
The variable a on the right side of this equation is acceleration, which is the second derivative
of position. Usually the force F depends on position as well as perhaps velocity, which means
that Newton’s second law is really a second-order differential equation.
For example, consider a mass hanging from a stretched spring. The force on such a mass
is proportional to the position y, i.e.
F = −ky,
where k is a constant. Plugging this into Newton’s second law gives the equation
−ky = my′′
.
The solutions to this differential equation involve sines and cosines, which is why a mass
hanging from a spring will oscillate up and down. Similar differential equations can be used
to model the motion of a pendulum, the vibrations of atoms in a covalent bond, and the
oscillations of an electric circuit made from a capacitor and an inductor.
It follows that k = 0.020/sec. Now, the general solution to the differential equation is
T = 20 + Ce−kt
and plugging in t = 0 gives
170 = 20 + C,
which means that C = 150 ◦
C. Thus
T = 20 + 150e−0.02t
.
To find how long it will take for the temperature to reach 30 ◦
C, we plug in 30 for T
and solve for t. The result is that t = 135 minutes .
The technique used in this example of substituting the initial conditions into the
differential equation itself is quite common. It can be used whenever the differential
equation itself involves an unknown constant, and we have information about both y(0)
and y′
(0).
EXERCISES
1. A sample of an unknown radioactive isotope initially weighs
5.00 g. One year later the mass has decreased to 4.27 g.
(a) How quickly is the mass of the isotope decreasing at
that time?
(b) What is the half life of the isotope?
2. A cell culture is growing exponentially with a doubling time
of 3.00 hours. If there are 5,000 cells initially, how long will
it take for the cell culture to grow to 30,000 cells?
3. During a certain chemical reaction, the concentration of
butyl chloride (C4H9Cl) obeys the rate equation
d[C4H9Cl]
dt
= −k[C4H9Cl],
where k = 0.1223/sec. How long will it take for this reaction
to consume 90% of the initial butyl chloride?
7. APPLICATIONS OF DIFFERENTIAL EQUATIONS 7
4. A capacitor with a capacitance of 5.0 coulombs/volt holds an
initial charge of 350 coulombs. The capacitor is attached to
a resistor with a resistance of 8.0 volt · sec/coulomb.
(a) How quickly will the charge held by the capacitor
initially decrease?
(b) How quickly will the charge be decreasing after
20 seconds?
5. A bottle of water with an initial temperature of 25 ◦
C is
placed in a refrigerator with an internal temperature of 5 ◦
C.
Given that the temperature of the water is 20 ◦
C ten
minutes after it is placed in the refrigerator, what will the
temperature of the water be after one hour?
6. In 1974, Stephen Hawking discovered that black holes emit a
small amount of radiation, causing them to slowly evaporate
over time. According to Hawking, the mass M of a black
hole obeys the differential equation
dM
dt
= −
k
M2
where k = 1.26 × 1023
kg3
/year.
(a) Use separation of variables to find the general solution
to this equation
(b) After a supernova, the remnant of a star collapses into a
black hole with an initial mass of 6.00 × 1031
kg. How
long will it take for this black hole to evaporate
completely?
7. According to the drag equation the velocity of an object
moving through a fluid can be modeled by the equation
dv
dt
= −kv2
where k is a constant.
(a) Find the general solution to this equation.
(b) An object moving through the water has an initial
velocity of 40 m/s. Two seconds later, the velocity has
decreased to 30 m/s. What will the velocity be after ten
seconds?
8. A population of bacteria is undergoing logistic growth, with
a maximum possible population of 100,000. Initially, the
bacteria colony has 5,000 members, and the population is
increasing at a rate of 400/minute.
(a) How large will the population be 30 minutes later?
(b) When will the population reach 80,000?
9. Water is being drained from a spout in the bottom of a
cylindrical tank. According to Torricelli’s law, the
volume V of water left in the tank obeys the differential
equation
dV
dt
= −k
√
V
where k is a constant.
(a) Use separation of variables to find the general solution
to this equation
(b) Suppose the tank initially holds 30.0 L of water, which
initially drains at a rate of 1.80 L/min. How long will it
take for tank to drain completely?
10. The Gompertz equation has been used to model the
growth of malignant tumors. The equation states that
dP
dt
= kP(ln Pmax − ln P)
where P is the population of cancer cells, and k and Pmax are
constants.
(a) Use separation of variables to find the general solution
to this equation.
(b) A tumor with 5000 cells is initially growing at a rate of
200 cells/day. Assuming the maximum size of the tumor
is Pmax = 100,000 cells, how large will the tumor be after
100 days?
8. Answers
1. (a) 0.67 g/yr (b) 4.39 years 2. 7.75 hours 3. 18.83 sec 4. (a) 8.75 coulombs/sec (b) 5.3 coulombs/sec
5. 8.6◦C 6. (a) M = 3
√
C −3kt (b) 5.71×1071 years 7. (a) v =
1
kt +C
(b) 15 m/s
8. (a) 39,697 (b) t = 51.4 min 9. (a) V =
1
4
(C −kt)2
(b) 33.3 min
10. (a) P = Pmax exp Ce−kt
. (b) 45,468 cells