Welcome to the web site of Dr. Mark P. Silverman, Jarvis Professor of Physics at Trinity College. I am a physicist whose scientific interests and contributions fall broadly in the areas of quantum physics, atomic & nuclear physics, optics & electromagnetism, and gravitation. I am also a teacher who has developed humane and effective educational methods for teaching science at all levels of instruction. At this site you can find books and papers in which I discuss my scientific research and educational experiments. More about me »

Motivating Students to Learn Science

“The science teacher whose instruction goes no further than the textbook, whose notes have yellowed with age from unvarying use, and whose concept of scientific experience is ritualised repetition of procedure cannot hope to motivate and inspire students. Teachers must, themselves, be motivated and inspired: to read avidly and regularly in order to learn lessons of the past and keep abreast of the present; to experiment and tinker in order to teach with confidence based on personal experience; to try in all ways possible to make the science classroom reflect accurately the attributes and activities of a place where real science is done.”

—M P Silverman, Motivating Students to Learn Science, Creative Education (2015) 6

Letter of Advice to the President from a Scientist

“Dear Mr President,

As an American atomic and nuclear physicist I was greatly heartened by your declaration to be guided by science in the implementation of public policy...In the course of your administration you may find that very real scientific facts run counter to what you consider self-evident truths. I will give you a critically important example that underlies nearly every major decision you are likely to make in the next four years. Let us start by considering climate change.”

—M. P. Silverman, Letter of Advice to the President from a Scientist Regarding Climate Change Policies (3 Feb 2021), sent to President Joseph Biden and to the Washington Post

New Books

And Yet It Moves: Strange Systems and Subtle Questions in Physics

2nd Edition (April 2015) Cambridge University Press - ISBN-13: 978-0521446310

And Yet It Moves cover

This book of essays, recently released as a second edition, is a largely nonmathematical account of some of the strange behaviour, both classical and quantum, exhibited by diverse physical systems. Drawn from the author's wide-ranging researches in quantum mechanics, atomic and nuclear physics, electromagnetism and optics, gravity, thermodynamics, and the physics of fluids, the essays describe different physical systems whose behaviour provokes surprise and challenges the imagination. Read more »

A Certain Uncertainty

Cambridge University Press 2014 - ISBN-13: 978-1107032811

A Certain Uncertainty cover

This book examines randomness, chance, and uncertainty in matters pertaining to fundamental questions of physics as well as to issues that are likely to affect readers in personal ways. Among the topics discussed are controversial issues concerning the decay of radioactive nuclei, predictability of the stock market, safety of commercial aircraft, accelerating rise in Earth’s temperature, interpretation of medical tests, detection of drug use to enhance athletic performance, and more. The book helps develop proficiency in the use of probability and statistics to understand a multiplicity of physical processes. Read more »

Recent Work


Crowdsourcing is a method of sampling by submitting a problem to a large number of diverse, independent, non-experts (the "crowd").  The underlying idea is that a crowd can collectively arrive at a better solution than a small group of experts.  This is not a mathematical theorem, but a hypothesis to be tested.  In these papers I analyse a crowdsourcing experiment to study the distribution, information content, and reliability of a crowd's response.

Part I: M P Silverman, "Crowdsourced Sampling of a Composite Random Variable: Analysis, Simulation, and Experimental Test", Open Journal of Statistics 9 (2019) 494-529.

Part II: M P Silverman, "Extraction of Information from Crowdsourcing: Experimental Test Employing Bayesian, Maximum Likelihood, and Maximum Entropy Methods", Open Journal of Statistics 9 (2019) 571-600.

Randomization of a Deck of Playing Cards

The question of how many shuffles randomise a pack of playing cards has fascinated mathematicians, scientists, and the general public. Different theoretical approaches have led to statistically different answers. This paper reports a comprehensive test of manual and mechanical shuffles examined in regard to runs, rising sequences, entropy & information loss, and other statistical measures.

M P Silverman, "Progressive Randomization of a Deck of Playing Cards: Experimental Test and Statistical Analysis of the Riffle Shuffle," Open Journal of Statistics 9 (2019) 268-298

Forces on a Ladder

Predicting the forces of reaction on a simple ladder has long remained an unresolved physical problem. The difficulty is that the system is statically indeterminate and therefore requires complementary information. Part I analyses three fundamental models of a ladder and reports a comprehensive experiment test that reveals the correct model. Part II examines the role of friction at the base and top supports of the ladder. Many thousands of accidents annually involve falls from ladders, and many of these cases are litigated in the courts. The solution to the ladder problem provides guidelines for safe use of ladders and a reliable theoretical basis for ascertaining whether collapsed support structures met statutory building codes.

Part I: M P Silverman, "Reaction Forces on a Fixed Ladder in Static Equilibrium: Analysis and Definitive Experimental Test of the Ladder Problem", World Journal Mechanics 8 (2018) 311-342 - Full paper

Part II: M P Silverman, "The Role of Friction in the Static Equilibrium of a Fixed Ladder: Theoretical Analysis and Experimental Test", World Journal of Mechanics 8 (2018) 445-463 - Full paper

Test of Elastica Theory

Numerous objects, either natural (e.g. trees) or designed (e.g. fishing rods), take the form of a flexible rod (cantilever). This paper presents a comprehensive analysis and experimental test of the exact theory of the bending of a tapered rod. The extent of flexure in these experiments far exceeded the range of applicability of approximations that linearize the theory or neglect the tapering of the rod.

See: M P Silverman and J Farrah, "Bending of a Tapered Rod: Modern Application and Experimental Test of Elastica Theory", World Journal Mechanics 8 (2018) 272-300 - Full paper

Brownian Motion of Radioactive Particles

Brownian motion can be analysed by either of two equivalent methods (Fokker-Planck equation or Langevin equation) if the motion is continuous. However, the diffusion of radioactive particles is not a continuous process, since their paths terminate when the particles decay. Nevertheless, this paper demonstrates that the two methods yield identical statistical information in the case of a constant nuclear decay rate (Rutherford-Soddy law).

See: M P Silverman and A Mudvari, "Brownian Motion of Radioactive Particles: Derivation and Monte Carlo Test of Spatial and Temporal Distributions", World Journal of Nuclear Science and Technology 8 (2018) 86-119

Brownian Motion of Decaying Particles

Methods to detect and monitor radioactive gases require understanding the combined effects of diffusion and decay. Derivation and solution of the Fokker-Planck and Langevin equations appropriate to these two processes reveal important differences in the spatial and temporal statistics of stable and unstable particles.

See: M P Silverman, "Brownian Motion of Decaying Particles: Transition Probability, Computer Simulation, and First-Passage Times", Journal of Modern Physics 8 (2017) 1809-1849

Residence Time of Radon in an Open Volume

Measurement of radon concentration by passive diffusion depends on an accurate assessment of how many atoms pass through the detection region in a given time. An alternative, but equivalent, micro-statistical approach is to determine the mean time spent by one atom in the detection region. The solution of this problem, besides validating the original macro-statistical approach, addressed a number of interesting issues concerning the random walk of a particle subject to random decay.

See: M P Silverman, “Analysis of Residence Time in the Measurement of Radon Activity by Passive Diffusion in an Open Volume: A Micro-Statistical Approach”, World Journal of Nuclear Science and Technology 7 (2017) 252-273

Search for Anomalies in the Decay of Mn-54

Mn-54 decays to Cr-54 by the capture of an orbital electron. Interactions that affect the inner electrons could conceivably affect the Mn-54 decay rate. In response to published claims that Mn-54 decay on Earth is influenced by the Sun, I performed a battery of statistical tests on a long record of Mn-54 decays. The outcome was an exquisite agreement with standard nuclear physics.

See: M P Silverman, "Search for Anomalies in the Decay of Radioactive Mn-54", Europhysics Letters 114 (2016) 62001

See: Editorial highlight by Hamish Johnston in Physics World:

New Method to Measure Indoor Radon Concentration

Radon is a radioactive gas that seeps into homes and workplaces and is the leading cause of lung cancer among non-smokers. Many methods exist to measure indoor radon concentration, but accurate ones require specialised equipment and long waiting times, since samples are sent to commercial labs for processing. I have developed a method that is accurate, uses the simplest, overall least expensive nuclear counters, and can be implemented in situ by those who wish to make their own measurements.

See: M P Silverman, "Method to Measure Indoor Radon Concentration in an Open Volume with Geiger-Mueller Counters: Analysis from First Principles", World Journal of Nuclear Science and Technology 6 (2016) 232-260

Search for Non-Standard Radioactive Decay

Reports of periodic variations in the decay rate of certain unstable nuclei are believed to violate known physical laws and have been disputed, but the effects are weak and the situation remains ambiguous. I propose alternative methods to search for non-standard radioactive decay that are much more sensitive to variations in the decay rate.

See: M P Silverman, "Search for Non-Standard Radioactive Decay Based on Distribution of Activities", Europhysics Letters 110 (2015) 52001

See: M P Silverman, Effects of a Periodic Decay Rate on the Statistics of Radioactive Decay: New Methods to Search for Violations of the Law of Radioactive Change, Journal of Modern Physics 6 (2015) 1533-1553

Probability Density vs Number of Sequential Samples

Cheating or Coincidence?

Correspondences between a student's solutions to a set of homework problems and the instructor's solutions may suggest that the student copied the work from an illicit source. A statistical analysis can help determine how likely it is that the points of similarity are coincidental. Since effects of randomness can be counter-intuitive and easily misjudged, the analysis must be done objectively and accurately to reduce the chance of a false accusation.

See: M P Silverman, "Cheating or Coincidence? Statistical Method Employing the Principle of Maximum Entropy for Judging Whether a Student Has Committed Plagiarism", Open Journal of Statistics 5 (2015) 143-157

Probability vs Class Size

The Half-Life of Radioactive Nuclei

Estimating the half-life of an unstable nucleus from just two measurements does not give an accurate result. However, a histogram of a large number of two-point estimates leads to a nearly perfect Cauchy distribution centred at the true value.

See: M P Silverman, “Theory of nuclear half-life determination by statistical sampling”, Europhysics Letters 105 (2014) 22001 p1-p5. doi: 10.1209/0295-5075/105/22001

Probability of Half-Life Estimate

Rising Temperatures and Urban Heat Stress

The Earth’s surface temperature is increasing. However, measurements of subterranean temperature variations, which are largely insensitive to the vicissitudes of weather, indicate that the rate of temperature rise in cities can far outpace the rate of global or regional warming.

See: M P Silverman, “Statistical analysis of subsurface diffusion of solar energy with implications for urban heat stress”, Journal of Modern Physics, 5 (2014) 751-762

Variation of Temperature with Time and Depth

Recurrent Patterns in Random Events

The common perception (see Wikipedia) of randomness is that it is devoid of all order, pattern, or predictability. However, that is a misperception. It is precisely the existence of patterns of recurrence that characterise a random process and serve usefully in many ways, among which is to test the randomness of quantum events.

See: M P Silverman, “Numerical Procedures for Calculating the Probabilities of Recurrent Runs”, Open Journal of Statistics 4 (2014) 122-131

Variation of Temperature with Time and Depth


I am available for lecturing on a variety of topics related to my research of which the following are some examples. Also, see Lectures for more details.

  • The strange behaviour of quantum particles
  • New directions in electron microscopy and interferometry
  • Do radioactive nuclei decay randomly?
  • Seeing through turbid media with polarised light
  • Quantum stabilisation of stellar black holes
  • Crowdsourcing: A radical approach to problem solving
  • A warning from the weather under ground: Solar energy diffusion and urban heat stress
  • The Ladder Problem: Physics in the Service of Safety and Justice


A Certain Uncertainty And Yet It Moves And Yet It Moves - 2nd Edition More Than One Mystery Waves and Grains Probing The Atom A Universe of Atoms Quantum Superposition