Justin Ivins
Professor Rowell
Math 220
5 April 2014
Statistical Mechanics
Statistical mechanics is a very broad subject with many other concepts under its umbrella. This topic has entire classes dedicated to it, with hundreds of theories and equations, so instead of unrealistically trying to master a whole course I instead sought to get a general understanding of the topic to the point that I could apply what I learned to future courses featuring statistical mechanics. However, to understand the topic I obviously had to go in depth into the main points of the field, and examine the contributions that pushed the field forward. After my initial research I found that the topic of statistical mechanics is like the label on a tool box with dozens
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Classical mechanics explores the movement of bodies because of other sources, and can determine future and past locations of molecules at any time if given the location and velocity of a molecule at any one time. Classical can be very accurate if given large enough objects and the speed of an object is not near the speed of light (Leland). While classical deals with objects on a human scale, quantum mechanics deals with objects on an atomic scale. Quantum mechanics throws away the idea that we can define where a molecule’s trajectory is heading, so we must instead take probabilities and average the changing position over time (Leland). All of this is just the basic workings of physical mechanics without statistical mechanics added in it yet. Statistical mechanics combines the principles of statistics with the laws of classical and quantum mechanics, particularly with thermodynamics (Leland). Statistics gets added in with the idea of statistical ensembles, which are probability distributions over all possible states of the system. In quantum mechanics, statistical mechanics introduces stat ensembles that use a large collection in the system and creates a probability distribution over all the possible systems (Jackson). In classical mechanics, statistical mechanics uses a probability distribution over phase points as the ensemble …show more content…
Statistical thermodynamics is a branch of physics that applies probability theory to the study of the thermodynamic behavior of a large number of particles (Laurendeau). Statistical mechanics provides a blueprint for relating the microscopic properties of singular atoms and molecules to the macroscopic overall properties of materials that can be observed physically (Laurendeau). This relationship explains thermodynamics. Statistical mechanics provides an atom-sized interpretation of macroscopic thermodynamic characteristics such as heat, entropy, and work. It enables the thermodynamic properties of the whole object to be related to the microscopic data of individual atoms or molecules (Laurendeau). This ability to make macroscopic calculations based on microscopic characteristics is the main benefit of statistical mechanics over classical thermodynamics. Josiah Willard Gibbs used these definitions to explain the laws of thermodynamics as consequences of the statistical properties of large ensembles of particles, and officially created the field of statistical mechanics. His book Elementary Principles in Statistical Mechanics is considered the foundation of the field, and for the first time documented how quantum-like laws could arise from the old classical mechanical system
The math/science program at PWC Governor’s School presents challenging learning opportunities and well as motivated classmates and professors that can further my curiosity and devotion for the mathematical and scientific world around me. It also withholds a window of opportunity for me to move closer in attaining my career aspiration. In return, I can provide collaboration, leadership, and most importantly, an intelligent mind capable of creating pioneering, innovative, and inspiring ideas. Together, the students/faculty of PWCS Governor’s School and I can collaborate to form the most ground-breaking solutions.
Thermodynamic equilibrium leads to the large-scale definition of temperature, as opposed to the small-scale definition related to the kinetic energy of the molecules. The first law of thermodynamics relates the various forms of kinetic and potential energy in a system to the work which a system can do and to the transfer of heat. This law is sometimes taken as a definition internal energy, and introduces an extra state variable, enthalpy. The first law of thermodynamics allows for many possible states of a system to exist. But experience indicates that only certain states occur. This leads to the second law of thermodynamics and contrast between another state variable called entropy. The second law stipulates that the total entropy of a system plus its environment can not decrease; it can remain constant for a reversible process but must always increase for an irreversible process. Thermal energy is the energy a substance or system has due to its temperature, i.e., the energy of moving or vibrating molecule. Thermodynamics involves measuring this energy, which can "exceedingly complicated," according to David McKee, a professor of physics at Missouri Southern State University. "The systems that we study in thermodynamics … consist of very large numbers of atoms or molecules interacting in complicated ways. But, if these systems meet the right criteria, which we call equilibrium, discovered with a very small number of measurements or
Quantum Mechanics is a branch of physics that describes the structure and behavior of matter.
For people who are not statisticians, they may wonder what statisticians do, and how statistics could be applied in daily life. Statistics: A Guide to the Unknown is a supplementary reading materials designed for general readers even if he or she did not learn enough knowledge of statistics, mathematics and probability. Besides, it could give statisticians a general understanding of the important role of statistics in society. This book also analyzes how statistics assists people to gain useful information from massive data sets. In order to form a more respected book, the editors invite many distinguished researchers in statistics as authors. The book consists of twenty-five essays from different fields, including public policy and social science, science and technology, biology and medicine, business and industry, and hobbies and recreation. Each essay provides readers a description of how statistical methods are applied to solve issues in that field.
I also learned that mathematics was more than merely an intellectual activity: it was a necessary tool for getting a grip on all sorts of problems in science and engineering. Without mathematics there is no progress. However, mathematics could also show its nasty face during periods in which problems that seemed so simple at first sight refused to be solved for a long time. Every math student will recognize these periods of frustration and helplessness.
Physics can be found in all aspects of our lives and the world around us including the activities in which we find the most enjoyment. They may not be noticeable to the naked eye or even to our senses but they are there and when we become familiar with the concepts of physics then we began to ‘see’ physics everywhere.
Ms. Kinney showed me the physics in everything from toys to poetry. We discussed Walt Whitman's "When I heard the Learn'd Astronomer" and Robert Frost's "Birches" along with Newton's Laws and Einstein's Theory of Relativity. She had an amazing ability to teach the most complicated concepts in physics without letting us lose sight of the simple wonder of it all. She made physics come alive for me.Ms. Kinney's class was one of the most rigorous and demanding courses that I have faced, but she gave me so much more than she asked of me. She taught me to love learning and physics, and to appreciate the magic and excitement of discovering and mastering new knowledge. Many students got so caught in the difficulty and high expectations of her class that they did not realize the opportunity we were being given.With Ms.
The zeroth law of thermodynamics states that “when two systems are each in thermal equilibrium with a third system, the first two systems are in thermal equilibrium with each other.” (Drake P.1). The first law of thermodynamics states that the change in internal energy of a system is equivalent to the total work done by the system subtracted from the total heat transfer into the system. This law is represented by the equation The variable represents the change in internal energy of the system, represents the total heat transferred into the system, and represents the total work done by the system. The second law states that heat flows spontaneously from hotter to colder regions but never in the reverse direction. It also states that the total entropy can never decrease over time for an isolated system; it will always increase over time. Additionally, the changes in entropy in the universe can never be negative. The third law states that “the entropy of a perfect crystal of an element in its most stable form tends to zero as the temperature approaches absolute zero.” (Drake P.1). Thermodynamics developed quickly throughout the 19th century because of the need to improve steam engines and how they worked. The thermodynamics laws can be applied to “all physical and biological systems” (Drake P.1). These laws of thermodynamics are able to give people an explanation about a variety of changes in the energy of a system, along with its
Ever since I was a child, I have had a great interest for the automotive industry. From car trivia to novel innovations, my innate passion for the automotive industry has always made me research the minutest detail of every vehicle that interested me. Since elementary school I would draw sketches of cars which incorporated technology which were unheard of at that time; novel devices such as electrochromic windshields, HUD displays, and wind turbines which would constantly re-generate electricity for the car. While growing up, my hobbies largely consisted of constructing countless Lego and Meccano sets, and repairing my mom’s 19 year-old car. In middle school, math and science were my favorite subjects: applying science and mathematics to solve real-world problems has fascinated me and I have also taken further steps to reach my goals. By the age of thirteen I devised a scaled model of a heliostat power plant, which successfully powered a light bulb. The mathematics beyond the focus points of parabolic dishes and thermodynamics was very advanced for my age, but I took up the challenge...
Statistics work for everything when there is a lot of it. They work for money, molecules, atoms, star systems, and even people. People tend to adhere to statistics when there is a fair amount of people to stifle the occasional fluctuations in human behavior. Many things we do depend on statistics. Take war for example. War is a very good example, since the outcome depends more on the general strategy of the whole war, than on individual soldiers. It follows definite rules that can be expressed in formulas. The individual people in war tend to become statistics, in the eyes of the high command, the public, as well as in their own perception. Tim O’Brien wonderfully illustrates this in his essay “How to Tell a True War Story.” He relates that there is no point to any events or actions according to the perception of the soldier during a war. “You smile and think, ... what’s the point?” (469) he says. A person then becomes nothing more than a statistic -- a part of a whole behaving in a random way. If there is no point to existence, then his actions are truly random. Something truly random can be easily studied, stimulated, expressed in some numbers, percentages, probabilities. This randomness of the soldier is what the whole military apparatus depends on. Consider: if the life of a soldier during war had a point, if he realized that there is some underlying meaning, wouldn’t he strive toward the goal assigned by that meaning? He would, for that is in human nature. Now, if there was no meaning in his perception, he could easily be persuaded that a particular thing must be done. He will obediently follow.
Thermodynamics is the branch of science concerned with the nature of heat and its conversion to any form of energy. In thermodynamics, both the thermodynamic system and its environment are considered. A thermodynamic system, in general, is defined by its volume, pressure, temperature, and chemical make-up. In general, the environment will contain heat sources with unlimited heat capacity allowing it to give and receive heat without changing its temperature. Whenever the conditions change, the thermodynamic system will respond by changing its state; the temperature, volume, pressure, or chemical make-up will adjust accordingly in order to reach its original state of equilibrium. There are three laws of thermodynamics in which the changing system can follow in order to return to equilibrium.
Probability and Statistics most widespread use is in the arena of gambling. Gambling is big all over the world and lots of money is won and lost with their aid. In horse racing especially the statistics of a horse in terms of its physical condition and winning history sway numbers of persons into believing that the mathematical evidence that is derived can actually be a good indicator of a race’s outcome. Usually it is if the odds or probability are great in favor of the desired outcome. However the future is uncertain and races can turn out any of a number of different ways.
Between 1850 and 1900, the mathematics and physics fields began advancing. The advancements involved extremely arduous calculations and formulas that took a great deal of time when done manually.
Furthermore, my Physics class that introduced classical mechanics has prepared me for my major in science in mathematics. This class helped me because the professor focused on the general solution to a problem by using a
This evaluation has not only allowed me explore calculus more in depth, but also physics, and the way the world works. This has personally allowed me to explore the connections between math and real-world situations, which is hard to find in textbooks.