1. The Optimal Can an uncanny approach* Carlos Zambrano & Michael Campbell *spiced up with concepts from statistical mechanics… This talk is dedicated to Dr. Gerald Gannon for inspiring research that is deep and accessible. v 2014.02.26.00

2. naively, least cost from minimal surface area But cans in a store look more like this… what gives?

3. material cost formula Bulk of sheets of tin plate steel used to make cans * www.alibaba.com/product-gs/656173961/metal_food_can.html ** www.emirapackaging.com/english/ASTM-A624M-84.htm *** www.alibaba.com/product-gs/555207963/tinplate_for_two_piece_can.html

4. storage costs “A simple rule to keep in mind is the larger the quantity you have shipped at once, the less you will pay per container for freight.” * Rectangular Storage – consider two situations – Rectangular cabinets: real estate / rental costs – 18-Wheeler Cargo Trailers: local delivery costs is a flat fee $150 A few points about cans in rectangular storage – there will always be a necessary amount of unused volume because a can does not have a rectangular top and bottom (can opener requirement) – we can see below that the dimensions of the cargo trailer will result in some radii and heights being more efficient – we will only consider practical dimensions for cans (Table 1) * http://www.skolnik.com/faq.php

5. storage

6. storage possibilities (20’x8’x8’6”)

7. calculation of storage cost

8. maximum cans in storage space: the built-in discontinuity

9. total shipping cost per volume of canned goods

10. the model: material + storage/shipping cost

11. stat mech… phase transition? * Spring 2012, c.f., “The Heart of Fifth Avenue Shopping is Heading to the South”, N.Y. Times, 4 Sep. 2012

12. computational analysis

13. some final thoughts on phase transitions and symmetry breaking

14. epilog: phase transitions & symmetry breaking a phase transition is the transformation of a system from one state (of matter, existence, etc) to another – water freezing into ice or vaporizing to steam – liquid crystals aligning in the nematic phase, so they will work in an LCD television – the early universe cooling to break the electroweak symmetry, resulting in the electromagnetic and weak forces decoupling into two “distinct” forces phase transitions are characterized by a “non-analyticity” in a function called the “free energy” which measures the amount of work a (thermodynamic) system can perform – some examples of non-analytic indicators – a discontinuity in the: function, or its derivative, or its second derivative, …, or a function that’s infinitely differentiable with taylor series not matching the function (at one or more points)

15. mechanism of phase transitions an order parameter is a function of a physical parameter describing a property of the system which typically is zero in one phase and non-zero in another – the magnetization of a ferromagnetic system is an order parameter (function of temperature)which is zero above a certain temperature (no magnetic properties), and non- zero below that temperature (magnetic properties) – if there is a (theoretical) phase transition, it is the result of a non-analyticity in the free energy – a visual cue for a phase transition frequently (but not always) is a break or corner/cusp on the graph of the order parameter as a function of the physical parameter

16. graph indicative of a first-order phase transition first-order phase transition: liquid crystals * * Fig 2.8.1b, “Investigation of Parameters of Liquid Crystal Composite System”, 2nd ICWET 2011 Proc., Int. Jour. of Computer Appl. graph of light absorption versus temperature note the discontinuity at 110 degrees C as T increases, crystals change from a nematic (liquid-solid) state to an isotropic (liquid state) at the clearing point transition temperature (77 deg C here) which is also in fact discontinuous and first-order

17. graph indicative of a second-order phase transition second-order phase transition: 2D Ising ferromagnet* * simulation at http://www.ibiblio.org/e-notes/Perc/trans.htmhttp://www.ibiblio.org/e-notes/Perc/trans.htm graph of Spontaneous Magnetization (M) versus Temperature (T) note M is a continuous function of T and approaches T c ≈ 2.27 with infinite slope at T c, the derivative χ (susceptibilty) diverges and is discontinuous alignment of spins change from aligned (magnetic) state to a paramagnetic (nonmagnetic) state at the critical temperature T c this is continuous and 2nd-order behavior ferromagnetic phase low temperature broken symmetry – flipping all spins yields different result (spins down) paramagnetic phase high temperature symmetry – flipping all spins yields same (random directions)

18. phase order of optimal can

19. computational analysis of optimal can model

21. open problem