An Introduction to Brownian Motion

An Introduction to Brownian Motion Brownian motion is the random movement of particles in a fluid due to their collisions with other atoms or molecules. Brownian motion is also known as pedesis, which comes from the Greek word for leaping. Even though a particle may be large compared to the size of atoms and molecules in the surrounding medium, it can be moved by the impact with many tiny, fast-moving masses. Brownian motion may be considered a macroscopic (visible) picture of a particle influenced by many microscopic random effects. Brownian motion takes its name from the Scottish botanist Robert Brown, who observed pollen grains moving randomly in water. He described the motion in 1827 but was unable to explain it. While pedesis takes its name from Brown, he was not the first person to describe it. The Roman poet Lucretius describes the motion of dust particles around the year 60 B.C., which he used as evidence of atoms. The transport phenomenon remained unexplained until 1905 when Albert Einstein published a paper that explained the pollen was being moved by the water molecules in the liquid. As with Lucretius, Einsteins  explanation served as indirect evidence of the existence of atoms and molecules. At the turn of the 20th century, the existence of such tiny units of matter was only a theory. In 1908, Jean Perrin experimentally verified Einsteins hypothesis, which earned Perrin the 1926 Nobel Prize in Physics for his work on the discontinuous structure of matter. The mathematical description of Brownian motion is a relatively simple probability calculation, of importance not just in physics and chemistry, but also to describe other statistical phenomena. The first person to propose a mathematical model for Brownian motion was Thorvald N. Thiele in a paper on the least squares method that was published in 1880. A modern model is the Wiener process, named in honor of Norbert Wiener, who described the function of a continuous-time stochastic process. Brownian motion is considered a Gaussian process and a Markov process with continuous path occurring over continuous time. What Is Brownian Motion? Because the movements of atoms and molecules in a liquid and gas is random, over time, larger particles will disperse evenly throughout the medium. If there are two adjacent regions of matter and region A contains twice as many particles as region B, the probability that a particle will leave region A to enter region B is twice as high as the probability a particle will leave region B to enter A. Diffusion, the movement of particles from a region of higher to lower concentration, can be considered a macroscopic example of Brownian motion. Any factor that affects the movement of particles in a fluid impacts the rate of Brownian motion. For example, increased temperature, increased number of particles, small particle size, and low viscosity increase the rate of motion. Brownian Motion Examples Most examples of Brownian motion are transport processes that are affected by larger currents, yet also exhibit pedesis. Examples include: The motion of pollen grains on still waterMovement of dust motes in a room (although largely affected by air currents)Diffusion of pollutants in the airDiffusion of calcium through bonesMovement of holes of electrical charge in semiconductors Importance of Brownian Motion The initial importance of defining and describing Brownian motion was that it supported the modern atomic theory. Today, the mathematical models that describe Brownian motion are used in math, economics, engineering, physics, biology, chemistry, and a host of other disciplines. Brownian Motion Versus Motility It can be difficult to distinguish between a movement due to Brownian motion and movement due to other effects. In biology, for example, an observer needs to be able to tell whether a specimen is moving because it is motile (capable of movement on its own, perhaps due to cilia or flagella) or because it is subject to Brownian motion. Usually, its possible to differentiate between the processes because Brownian motion appears jerky, random, or like a vibration. True motility appears often as a path, or else the motion is twisting or turning in a specific direction. In microbiology, motility can be confirmed if a sample inoculated in a semisolid medium migrates away from a stab line. Source Jean Baptiste Perrin - Facts. NobelPrize.org, Nobel Media AB 2019, July 6, 2019.

The Complete Guide to AP Human Geography FRQs

The Complete Guide to AP Human Geography FRQs SAT / ACT Prep Online Guides and Tips The free-response section is the hardest part of any AP test. Although AP Human Geography is much more straightforward than some other humanities AP exams, if you don't have your stuff down, the free-response section can still hit you where it hurts. In this article, I'll take you through the structure of the free-response section, some tips for answering AP Human Geography FRQsstrategically, examples of what actual questions look like, and a few places where you can find the best practice materials. AP Human Geography Free-Response Section Format There are three questions on the free-response section, each worth the same percentage of your score.You’ll have an hour and 15 minutes to answer all three questions.Free-response prompts will ask you to: Demonstrate an understanding of geographic models Analyze and evaluate geographic concepts Cite and explain examples of various geographic processes Synthesize different topics covered in the curriculum Most free-response questions have multiple parts, and you can earn anywhere from 6-10 raw points on each of them.It’s usually fairly obvious where the points are earned on these questions, as you’ll see in the examples in the upcoming sections. How to Answer AP Human Geography FRQs Before we get into examples with answer explanations, I’ll give you some essential steps for answering these types of questions.AP Human Geography free-response questions are typically pretty straightforward so you can attack them in a methodical fashion. Here's a sample question from the 2015 exam that I'll use as a reference so you can see how the different steps apply: Step 1: Read the Introductory Statement Before you start in on the first part of the question, make sure you read the short introductory blurb.It sets up the topic you’ll be analyzing and gives you firmer ground to stand on when addressing the rest of the question. In the sample question above, the introductory statement makes it clear that the question will deal with refugees and the political and social origins of mass emigration. Reading all the introductory statements for the three free-response questions in the section before you start answering any of them can help you decide where to begin.If you see a topic that is especially familiar, you might go for that question first because you’ll be able to answer it the fastest. Step 2: Identify the Key Command In each part of the question, underline what it’s asking you to do.Common examples of key commands include â€Å"compare,† â€Å"describe,† â€Å"identify,† and â€Å"explain.†It’s helpful to underline these words to keep yourself on track when responding to the question.This is also useful because it will get you in the habit of paying closer attention to the differences between each of these tasks. For example, a question that asks you to â€Å"identify† something will merit a much more succinct answer than one that asks you to â€Å"describe† it.In part A of the sample question, the key command is "define," meaning you could just give a one sentence answer that clearly outlines what a refugee is. For part B, the key command is "discuss," which denotes a longer answer that elaborates on each reason for refugees leaving their countries of origin. In part C, the key commands are "identify and explain," which would mean a very succinct answer followed by greater detail. The final part of the question asks you to "explain" two economic impacts, which means a couple of sentences of further clarification for each example you give.You’ll save time and earn points if you’re careful to make these distinctions! Step 3: Hit the Points Now it’s time to answer the question.Make an estimate of how many points are possible in this part of the question.For example, if a question tells you to give two examples of how a concept applies to a certain country’s economic landscape, there are likely two points that you can potentially earn for that part of the question.In the sample question, you can see relatively clearly that part A is worth one point, part B is worth three points, part C is worth two points, and part D is worth two points. Part C is the only tricky part - notice that you're only asked for one reason, but you must identify AND explain it. There is typically one point available for completing each key command.After making this determination, write a clear answer that addresses all of the points thoroughly and will make it easy for the grader to find your responses. Step 4: Reread and Double Check When you’ve finished writing your answer, reread the question and your response to make sure you’ve done everything it asked you to do.If you’re satisfied, move onto the next part of the question, and repeat the process of identifying key commands and hitting all the possible points.After you finish the last part, check over all of your answers for that question one last time to ensure that everything is the way you want it to be. Then you can move onto the next free-response question. Review your answers with a second, more critical eye. I think this picture is photoshopped, but I don't know for sure because I don't feel like Googling "can you have two pupil/iris combos within one eyeball?" Just kidding, I did Googleit, and it's probably not a thing, but not even the internet knows for sure.OoOoOoOo. AP Human Geography FRQ Examples In this section, I'll go through the answers to two different free-response questions that were asked on real AP Human Geography tests within the last couple of years. Here's the first question: For this question, a knowledge of the most prevalent geographic models and theories is very important.You need to be able to relate one of these models to the economic development of a specific country.This question requires almost all the skills listed in the first section of this guide. As you may have guessed, there were three points possible for the answer to the first part.Here are some examples of differences between the two models that you could identify and compare: Rostow’s five-stage model says that national economies are developing forward over time (never moving backward) while Wallerstein’s core-periphery model says that countries are static or can move backward in development. Rostow’s model says that international trade may help countries to grow economically while Wallerstein’s model says that international trade strengthens development in core countries and constrains development in peripheral countries. Rostow’s model provides a national-level analysis while Wallerstein’s model provides an international-level analysis. For part B, you need to show that you can apply these models in the context of Mexico and Brazil, countries that are in the midst of fully developing and modernizing their economies.One point would be earned for identifying a stage or part of one of the models, and a second point would be earned for explaining that stage or part.For example, you could talk about Stage 4, or the Drive to Maturity, in Rostow’s model.This stage is exemplified by increased manufacturing specialization and integration into the global economy. For part C, you would have to identify two examples of the core-periphery concept below the national level.One example might be a core city and its peripheral suburbs.Another example might be a core productive area surrounded by a less developed or impoverished periphery. Now, let’s take a look at a second example with an image component: Forpart A, we have to consult the map and determine one characteristic that the shaded countries have in common.Examples of characteristics you could mention include: They’re all developing countries. They’re former colonies. They all use plantation or small-scale agriculture. They’re all equatorial countries. Part A was worth one point.Notice that this only requires a very broad understanding of the nature of these countries or a basic knowledge of climate in different parts of the world. For part B, you could earn two points for identifying each impact of coffee farming.Possibilities include: Economic development:Coffee farming leads to increased employment, growth in GNP, development of infrastructure, improved foreign exchange, increased global trade and better international relations. Environmental impacts:Coffee farming causes harmful effects from agricultural chemicals, water use issues, deforestation, biodiversity loss, and soil erosion. You could also talk about how coffee farming might lead to too much economic dependence on a single commodity and cause the land to be used for commercial agriculture rather than food. Part C was also worth two points, one for identifying a way that increased consumption affects coffee-producing countries and one for explaining the effect in more detail.For example, you might say that increased consumption leads to increased production.This, in turn, means more resources dedicated to coffee production, the adoption of new technologies related to the industry, increased profit, and the introduction of new producers into the coffee industry. Finally, for part D, you have to explain a specifically urban change in the developed world related to coffee production.You might say that more coffee shops will start to spring up in urban areas.That gets you one point.Then, you could earn a second point if you got into a more specific discussion about how the placement of coffee shops is related to geographic theories.You might explain it in the context of consumer proximity, central place theory, and diffusion. How to Practice AP Human Geography FRQs You can find plenty of sample AP Human Geography free-response questions (and their corresponding answer guidelines) online.The College Board site has real test questions that were administered between 2004 and 2015 with sample responses and scoring guidelines.This is the best resource for free response questions because you can be sure that they’re accurate representations of what you’ll see on your exam in terms of content and difficulty level.There’s nothing wrong with using free-response questions crafted by test prep companies for preliminary practice, but you should always incorporate real questions into your studying at frequent intervals. Barron’s has a free practice test that you can take in timed or practice mode depending on how serious you’re feeling.It might be good to make use of the timed mode when you’re close to the real test so you can get an accurate feel for the conditions.It has three free-response questions just like the actual AP test, and you can consult scoring guidelines to check your answers (the multiple-choice portion of the test is scored automatically, but you have to do a little more work for free response).If you’re not averse to spending a bit of money ($25), you might also register with Learnerator to get access to a bunch of additional AP Human Geography practice free-response questions. It's time to fly free! Go, respond. It is your destiny. What's Next? If you want an overview of the whole exam with examples of multiple-choice and free-response questions, take a look at my survey of the AP Human Geography test including study tips and sample questions. Looking for more resources to use in preparing for this test? Check out my ultimate study guide for AP Human Geography! If you want more free response practice, you might consider getting a review book to supplement the online resources listed in this article. Here's a list of the best review books for AP Human Geography.